· abstract we study the mean curvature ow of closed curves in an euclidean space and de ne weak...

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Doctoral Thesis

Curve Shortening Flow in Higher Dimension

Author(s): Hättenschweiler, Jörg

Publication Date: 2015

Permanent Link: https://doi.org/10.3929/ethz-a-010412094

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DISS. ETH NO. 21708

Curve Shortening Flow in Higher Dimension

A dissertation submitted toETH Zurich

for the degree ofDoctor of Sciences

presented byJorg Hattenschweiler

Dipl. Math. ETH Zurichborn November 30, 1974

citizen ofRomanshorn TG, Switzerland

accepted on the recommendation ofProf. Dr. Tom Ilmanen, examinerPD Dr. Felix Schulze, co-examiner

2015

Abstract

We study the mean curvature flow of closed curves in an Euclidean space anddefine weak solutions that extend this flow beyond singular times. Our mainresult is the existence of such weak solutions until an extinction time, wherethe flow disappears, and uniqueness in a weak sense, which allows for certainre-parametrizations.

A weak solution consists of an open full measure set of regular times and afamily of closed Lipschitz curves depending on the time parameter. We char-acterize weak solutions by the following three properties: firstly the pullbackRadon measures are monotonically non-increasing in time, secondly the familyis Holder 1/2 and Lipschitz continuous away from the start time with respect tothe strip pseudo-distance, and thirdly the family satisfies, up to parametrization,the smooth mean curvature equation in each connected set of regular times.

The strip pseudo-distance, which occurs in the second criterion, gives rise toa metric on the space of Lipschitz curves modulo hairs and parametrizations. Ahair is a connected part of a curve, where it turns and takes the same way back.An important feature of the strip pseudo-distance, which is crucial for the proofof uniqueness, is that the strip pseudo-distance between two weak solutions is anon-increasing function of time.

To prove existence up to the extinction time we define a wave approximation,which is an alternative to the ramp approximation of Altschuler and Grayson.This wave approximation has the property that its projection to certain two di-mensional subspaces is a convex curve. We show that this property is perseveredunder the flow and that the limit at the first singular time is contained in someline. This implies that the solution, which arises from the wave approximation,is defined until the flow disappears.

To show convergence of a smooth approximation, we study curves whichlocally can be written as graphs of a Lipschitz function. Such a local Lipschitzgraph estimate is preserved under the flow for a short time. For two local Lips-chitz graphs a small strip pseudo-distance implies that the curves are Lipschitzclose. Among other things, we prove a lower bound for disk-type surfaces whichare bounded by local Lipschitz graphs.

To formulate the uniqueness, we define equivalence classes of weak solutions.We prove that the set of regular times only depends on the equivalence class. Alocal regularity theorem together with a covering argument shows that the setof singular times has Hausdorff dimension less or equal than 1/2. As a furtherconsequence of uniqueness, the total absolute curvature of a weak solution is anon-increasing function of time.

Each solution from a smooth approximation has a unique left continuousextension to singular times. We show that such an extension is continuous in aco-countable set of times and at each time the number of singular space-pointsis bounded by a multiple of the total absolute curvature.

i

Zusammenfassung

In dieser Arbeit untersuchen wir den mittleren Krummungsfluss von geschlosse-nen Kurven in einem euklidischen Raum. Wir definieren schwache Losungen,welche den Fluss uber Singularitaten hinaus erweitern. Unser Hauptresul-tat ist die Existenz solcher schwachen Losungen bis der Fluss komplett ver-schwindet und die Eindeutigkeit der Losungen in einem schwachen Sinn, inwelchem gewisse Umparametrisierungen erlaubt sind.

Eine schwache Losung besteht aus einer offenen Menge von regularen Zeiten,welche volles Mass hat und einer Familie von geschlossenen Lipschitz Kurven dievom Zeitparameter abhangen. Wir charakterisieren schwache Losungen durchdrei Eigenschaften: erstens sollen das pullback Radon Mass der Familie in derZeit nicht steigend sein, zweitens soll die Familie Holder 1/2 und weg von derStartzeit Lipschitz stetig bezuglich der Strip Pseudo-Distanz sein und drittenssoll die Familie in jeder zusammenhangenden Menge regularer Zeiten, bis aufParametrisierung, eine glatte Losung des mittleren Krummungsflusses sein.

Wir zeigen, dass die im zweiten Punkt vorkommende Strip Pseudo-Distanzeine Metrik auf dem Raum der Lipschitz stetigen Kurven modulo Haare undParametrisierung induziert. Ein Haar ist ein zusammenhangender Teil einerKurve, in dem die Kurve umdreht und den gleichen Weg zuruck lauft. Eine furdie Eindeutigkeit wichtige Eigenschaft ist, dass die Strip Pseudo-Distanz zweierschwachen Losungen eine in der Zeit nicht zunehmende Grosse ist.

Fur die Existenz definieren wir eine Wellen-Approximation, die eine Alter-native zu der Rampen-Approximation von Altschuler und Grayson darstellt.Diese hat die Eigenschaft, dass ihre Projektion auf gewisse zweidimensionaleUnterraume eine konvexe Kurve ist. Wir zeigen, dass diese Konvexitat unterdem Fluss erhalten bleibt und die Kurve im Grenzwert zu der ersten singularenZeit in einer Geraden enthalten ist. Dies sorgt dafur, dass die Losung, die ausder Approximation stammt, existiert bis der Fluss verschwindet.

Um die Konvergenz einer glatten Approximation zu beweisen, untersuchenwir Kurven, die lokal als Lipschitz Graphen geschrieben werden konnen. Dieselokale Lipschitz Graph Eigenschaft wird fur eine kurze Zeit unter dem Flusserhalten. Fur zwei lokale Lipschitz Graphen impliziert eine kleine Strip Pseudo-Distanz, dass die Kurven Lipschitz nahe sind. Unter andrem beweisen wirhier eine untere Schranke fur Disk-Typ Flachen, welche von lokalen LipschitzGraphen berandet werden.

Um die Eindeutigkeit zu formulieren, definieren wir Aquivalenzklassen vonschwachen Losungen. Wir zeigen, dass die Menge der regularen Zeiten nichtvom Reprasentanten abhangt. Ein lokales Regularitatstheorem zusammen miteinem Uberdeckungsargument zeigt, dass die Menge der singularen Zeiten eineHausdorff Dimension kleiner oder gleich 1/2 besitzt. Eine weitere Folge derEindeutigkeit ist, dass die totale absolute Krummung einer schwache Losungeneine monoton nicht wachsende Funktion der Zeit ist.

Losungen die aus einer glatten Approximation stammt konnen links stetigauf singularen Zeiten erweitert werden. Wie zeigen, dass eine solche Erweiterungin einer ko-abzahlbaren Menge von Zeiten stetig ist und zu jeder Zeit die Anzahlsingularer Raumpunkte durch ein vielfaches der totalen absoluten Krummungbeschrankt ist.

ii

Contents

1 Introduction 1

2 Mean Curvature Flow 72.1 Preliminaries and Definition . . . . . . . . . . . . . . . . . . . . . 72.2 First Variation Formula . . . . . . . . . . . . . . . . . . . . . . . 82.3 Tangential Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Short Time Existence . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Curve Shortening Flow 113.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . 123.2 Estimates for Higher Derivatives . . . . . . . . . . . . . . . . . . 163.3 Intersection Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Convergence at the First Singular Time . . . . . . . . . . . . . . 283.5 Integral Bounds for the Curvature . . . . . . . . . . . . . . . . . 313.6 Graph Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7 Local Lipschitz Graphs . . . . . . . . . . . . . . . . . . . . . . . . 433.8 Angenent’s Regularity Criterion . . . . . . . . . . . . . . . . . . . 613.9 Convex Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 683.10 Local Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Strip Pseudo-Distance 854.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . 864.2 Monotonicity of the Strip Pseudo-Distance . . . . . . . . . . . . . 1034.3 A Lower Area Bound . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4 Local Lipschitz Graph Estimate . . . . . . . . . . . . . . . . . . . 115

5 Weak Solution 1215.1 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . 1225.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3 A Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 1275.4 Extension to Singular Times and Regularity . . . . . . . . . . . . 1335.5 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.6 Further Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Appendices 149

A Ramps 149

B Minimal Surfaces 151

C Injective Approximation 153

D Monotonicity Formula 157

References 159

Notation Index 162

Curriculum Vitae 164

iii

1 Introduction

Let P be Riemannian manifolds and M be closed manifold. A solution of themean curvature flow is a smooth family F : M × [0, T ) → P of immersionssatisfying the equation

∂

∂tF (p, t) = H(p, t), (p, t) ∈M × [0, T ),

where H(p, t) is the vector valid mean curvature of F (M, t) at the point F (p, t).Must research are done in the case of hypersurfaces in an Euclidean space,

i.e. for co-dimension one immersions F (·, t) : Mn → Rn+1, and in particular forconvex surfaces and graphs. In this work we study only the one dimension caseof curves, but in an arbitrary dimensional Euclidean space. For curves the meancurvature H is simply the curvature k and the mean curvature flow is calledthe curve shortening flow. To emphasize this we denote the smooth family ofimmersions by x : S1 × [0, T ) → Rn, which satisfy the curve shortening flowequation

∂

∂tx(u, t) = k(u, t), (u, t) ∈ S1 × [0, T ),

where k(u, t) is the curvature of the curve γt = x(S1, t) at the point x(u, t).Clearly, most results are available for the curve shortening flow in R2 and wesummarize some of them. Convex planar curves are studied by Gage in [25] andGage and Hamilton in [26], where they show that the smooth flow of a convexcurve in R2 exists until the curve shrinks to a round point. In [28], Graysonproves that simple closed curve in R2 becomes convex before the curvature blowsup. This fact was simplified by Huisken [36], where he proves that a lower boundof the ratio between the extrinsic and intrinsic distance is preserved.

It is well known that the mean curvature flow of a compact manifold getssingular in finite time and it is a natural question if and how the flow can becontinued beyond singularities. There already exists many weak formulation ofthe mean curvature flow. We mention here the formulation with varifolds fromBrakke [9] and the level-set description with viscosity solutions from Evans &Spruck [22, 23] and Y.G. Chen, Y. Giga & S. Goto [10]. In [3], Ambrosio andSoner extend the level-set flow to higher co-dimension. These descriptions withvarifolds and sets are very general, but the uniqueness fails in several situationsand especially in higher co-dimension.

Figure 1: Non-Uniqueness

1

In co-dimension one, the mean curvature flow satisfy the avoidance principle,that two disjoint flow stays disjoint, provided one of them is compact. In [38],Ilmanen uses this property to define set-theoretic subsolutions by the compari-son with smooth compact flows. With this subsolutions he gives an alternativecharacterization of the level-set flow and shows a connection to Brakke’s solu-tion.

Related to the avoidance principle, in co-dimension one, the mean curvatureflow of an embedded hypersurface stays embedded. The avoidance principledoes not hold in higher co-dimension, and in this case the mean curvature flowis not expected to stay embedded. This leads to non-uniqueness of the level-setflow, as shown in figure 1 which we fund in [38]. To achieve a unique solution, itis necessary that one can distinguish between two touching circles and a figureeight, which is not possible if the solution is described by a set, a varifold or acurrrent.

Our goal is to define a weak solution of the curve shortening flow, which isbased on a parametrization x : S1 × [0, T )→ Rn and admits some uniqueness.

For the curve shortening flow in R2, the existence of such a weak solution isalready proven. In [5, 6], Angenent studied a general flow in a two dimensionalsurface P . This generalized flow includes the curve shortening flow. Angenentworks in the space Ω(P ) of equivalence classes of C1 immersions x : S1 → P ,modulo orientation-preserving C1 diffeomorphisms on S1. He proves that thenumber of singular points at the first singular time is finite. Moreover, he showsthat the limit curve at the first singular time can be reduced such that it isregular enough to restart the flow with this reduced initial curve. The reductionof the limit curve removes certain redundant part of the curve. We will latercome back to this point. He also proves that at a singular time either thenumber of self-intersection decreases, or the total absolute curvature decreasesby at least π. This allows him to construct a unique weak solution by restartingthe flow finitely many times.

Angenent proves: For any x0 ∈ Ω(P ) there is a generalized solution whichexists for all time, or else there is a generalized solution which has an emptyreduced limit curve at the final time. In the case P = R2 the solution disappearsin finite time, and hence there exists a singular time where the reduced limit isempty. Later Altschuler and Grayson [8] induce a ramp approximation, whichallows them to obtain the same unique weak solution in one step. This weaksolution is unique in Ω(R2) and smooth outside finitely many singular space-time points. Moreover, the number of the singular space-time points is boundedby a constant that only depends on the number of self-intersection and the totalabsolute curvature of the initial curve x0.

In higher co-dimension, it is an open question if the set of singular timesis finite, or not. Hence even we would have short time existence for a reducedlimit curve, it would not be possible to define a weak solution by a restartingprocess. We will obtain our weak solution as the limit of a wave approximation,that has some analogy to the ramp approximation of Altschuler and Grayson.Compared to the papers [5, 6] and [8], we will work with the normal motion andavoid the space Ω(Rn) as far as we can.

In [11], Deckelnick defines a weak solution for the equation xt = xuu/|xu|2,for x : S1 × [0, T ) → Rn, in a weighted L2 sense. Note that this equationdiffers only by a tangential motion from the curve shortening flow. His work isrelated, since his weak solution lies in the space x, xu ∈ L∞(S1 × [0,∞)) and

2

xuu ∈ L2(S1×[0,∞)), and hence defines a family of Lipschitz maps x(·, t). In hisexistence proof, Deckelnick solves the regularized problem xt = xuu/(|xu|2+ε2),which corresponds to a ramp solution. In the second paper [12], Deckelnickproves that the Hausdorff dimension of the set of singular times, for the solutionsthat arises as limit of the regularized problem, is less or equal than 1/2. Notethat Deckelnick’s solutions are not known to be unique and he only can provethis partial regularity for limits of the regularized problem. We will improvethis by our uniqueness result.

Before we can define our weak solutions, we have to induce the disk-sizeand strip pseudo-distance. The disk-size D0(x) of a closed Lipschitz curve isthe infimum of the area of disk type surfaces bounded by x. For two closedLipschitz curve x, y : S1 → Rn the strip pseudo-distance D0(x, y) is defined asthe infimum of the area of annulus type surfaces bounded by x and y. Thesequantities are of interest, since both are monotonically non-increasing as thecurve evolves by the curve shortening flow. This was first proven by Altschulerand Grayson [8], for the disk-size of space curves. Later Perelman [49] used thisidea for the curve shortening flow in a compact manifold (M, g(t)) which evolvesunder the Ricci flow.

It is amazing that there is not more research done about these quantities.We will study the disk-size and strip pseudo-distance in the space of Lipschitzmaps x : S1 → Rn and prove that the strip pseudo-distance is a pseudo-metric.By metric identification the strip pseudo-distance induces a metric in the spaceof Lipschitz curves modulo hairs and parametrization, where a hair of a curvex : S1 → Rn is a open connected interval I ⊂ S1 such that the restrictionx|I is a closed curve and D0(x|I) = 0. Moreover, we will prove that in eachequivalence class there exists a hairless representative, which is unique up toparametrization. Angenent’s reduced limit curve corresponds to this hairlessrepresentative. Hence Angenent’s generalized solution exists in Ω(R2) until thelimit curve has zero disk-size.

Note that it is an open question if hairs occurs in the curve shortening flow.However, the strip pseudo-distance helps us to study the flow in the present ofhairs and allows us to define a weak solution by the following three properties.

Definition. A weak solution on S1× [0, T ] consists of an open full measure setU ⊂ (0, T ) and a family of Lipschitz curves x(·, t) : S1 → Rn for t ∈ U , suchthat

a) the family of the pullback Radon measures µt on S1 defined by

µt(ψ) :=

∫S1

ψ(u)|xu(u, t)|du, ψ ∈ C0(S1,R)

is non-increasing in t ∈ U , in the sense of Radon measures.

b) the family x(·, t), t ∈ U , is Holder 1/2 in U and Lipschitz continuous awayfrom zero with respect to the strip pseudo-distance.

c) for every connected component V of U there exist a smooth curve shorteningflow x : S1 × V → Rn and a non-decreasing, degree one Lipschitz mapφV : S1 → S1 such that

x(u, t) = x(φV (u), t), (u, t) ∈ S1 × V.

3

Moreover, the smooth flow x is required to be singular at t = supV , ifsupV < T .

Note that a weak solution defines a continuous map x ∈ C0(S1×U,Rn) thatnot is expected to have a continuous extension on S1× [0, T ]. This correspondsto the picture that at a singular time the limit curve may jump to a hairlessreduced limit curve. The continuity in the strip pseudo-distance together withthe monotonicity of the Radon measures compensates this in a weak way.

Our main results are the following existence and uniqueness theorems.

Theorem. For every immersed curve x0 ∈ C1(S1,Rn) there exists a weaksolution (x, U) on S1 × [0, T ] that is maximal in the sense

limt→Tt∈U

D0(x(·, t)) = 0,

and satisfies the initial condition

x0 = limt→0t∈U

x(·, t).

Theorem. Let (x, Ux), (y, Uy) be weak solutions on S1 × [0, T ] satisfying theinitial condition x0, y0 respectively. If D0(x0, y0) = 0, then there holds Ux = Uyand for every open interval V ⊂ Ux = Uy there exist non-decreasing, degreeone Lipschitz maps φV , ψV : S1 → S1, and a smooth curve shortening flowxV : S1 × V → Rn such that

x(u, t) = xV (φV (u), t), (u, t) ∈ S1 × V,y(u, t) = xV (ψV (u), t), (u, t) ∈ S1 × V.

In particular, this result implies uniqueness in Ω(Rn), but it may look sub-optimal. We like to argue why the non-decreasing, degree one Lipschitz mapsdepends on V , and why we are not able to prove x(·, t) = y(·, t) if the initialcurves satisfy x0 = y0.

At a singular time t∗ the flow may form a triply covered segment, as shownin figure 2. In this case the hair I ⊂ S1 is not unique. By the cancellation of ahair we simple set the limit curve x∗ constant on I. Clearly, the image of thereduced limit curve does not depend on the chosen interval I, but there is nounique reduced parametrization.

t

Figure 2: Non-Unique Cancellation

4

An other consequence of hairs is, that we have to distinguish between regularand past-regular points. Locally in the interior of a hair I the limit curve x∗can be smooth. In this case the flow is locally smooth in a past neighborhoodup to and including the singular time, but since the hair collapses instantly theflow is not even continuous at this point.

Definition. Let (x, U) be a weak solution on S1 × [0, T ]. A point (u0, t0) inS1 × [0, T ] is called a regular respectively past-regular, if there exist δ > 0 suchthat for J := (t0−δ, t0+δ) respectively J := (t0−δ, t0] there holds J ⊂ [0, T ], anopen set H ⊂ S1 with u0 ∈ H, a non-decreasing Lipschitz map φ : H → (0, 1),and a smooth curve shortening flow x : (0, 1)× J → Rn, such that

x(u, t) = x(φ(u), t), (u, t) ∈ H × (J ∩ U).

Let R by the set of regular points, and Rp be the set of past-regular points.A time t0 is called regular respectively past-regular if S1 × t0 ⊂ R respec-

tively S1 × t0 ⊂ Rp. We denote by R the set of regular times and by Rp theset of past-regular times.

We will prove that for every maximal weak solution (x, U) there holds

U = R = Rp.

Intuitively this is true, since many point on a hair may be past-regular, but thereexists always a point where the hair is not smooth and this is a non past-regularpoint.

The weak solution (x, U) on S1×[0, T ] with initial condition x0 ∈ C1(S1,Rn)that arises from our existence proof, as limit of smooth solutions, satisfy moreregularity than general weak solutions. We will show that such a weak solutioncan be extended to singular times, by proving that for all t0 ∈ (0, T ] there existsthe left limit

x+(u, t0) := limt→t0

t∈U∩[0,t0]

x(u, t).

This allows us to define the space-time map X(u, t) := (x+(u, t), t) on S1×[0, T ]and to study the past-singular space-time points Σp := X(S1× [0, T ] \Rp). Wewill show that its parabolic Hausdorff measure is bounded by

H1;1/2(Σp) ≤ CL(x0),

where L(x0) is the length of the initial curve and C is constant depending onlyon the maximal area ratio of the initial curve x0. By our uniqueness theoremthis implies

Theorem. Let (x, U) be a weak solution on S1 × [0, T ] with initial conditionx0 ∈ C1(S1,Rn). Then there holds H1/2([0, T ] \ U) ≤ CL(x0).

Again for the weak solution that arisen as limit of smooth solutions, we showthat at each time t∗ ∈ (0, T ] the set of past-singular points in S1×t∗ consistsof finitely many closed intervals I1, ..., IN on which the map x+(·, t∗) is con-stant, i.e. there exists finitely many past-singular space points pi = x+(Ii, t∗).Moreover, we show that their number is bounded by

N ≤ 1

cK+(t∗),

where K+(t∗) is the left limit of the total absolute curvature and c > 0 is aconstant depending on nothing. This improves a result from Angenent in [6].

5

2 Mean Curvature Flow

Even though this theses is about curves, we define the smooth mean curvatureflow and some of its basic properties for general immersed submanifolds.

2.1 Preliminaries and Definition

Let M be a smooth m-dimensional manifold and F : M → P a smooth immer-sion in a n-dimensional Riemannian manifold (P, g). Usually we will considerP = Rn with the Euclidean metric. Let g be the pull-back metric of g on M .Let xi : U → Rm resp. yα : V → Rn be local charts in a neighborhood ofp ∈ U ⊂ M resp. q = F (p) ∈ V ⊂ P . Then ∂/∂xi is a basis of TpM , and dxi

spans TpM∗. We will use Latin indexes for i, j, k = 1, ...,m and Greek indexes

for α, β, γ = 1, ..., n. In these coordinates the metric g is given by

gij = g

(∂F

∂xi,∂F

∂xj

)= gα,β

∂Fα

∂xi

∂F β

∂xj.

Let ∇ = ∇g resp. ∇ = ∇g be the Levi Civita connection on TM resp. TP ,so that for a vector field X resp. X and a 1-form ω resp. ω on M resp. P ,

∇iXj =∂Xj

∂xi+ ΓjikX

k, ∇iωj =∂ωj∂xi− Γkijωk,

∇αXβ =∂Xβ

∂yα+ ΓβασX

σ, ∇αωβ =∂ωβ∂yα

− Γγαβωγ ,

where the Christoffel symbols Γijk and Γiβk are defined by

Γijk =1

2gil(∂gkl∂xj

+∂gjl∂xk

− ∂gjk∂xl

).

Γαβγ =1

2gασ

(∂gγσ∂xβ

+∂gβσ∂yγ

− ∂gβγ∂yσ

).

We denote by ⊥ resp. ‖ the orthogonal projection of TP to TM⊥ := TP/TMresp. TM . Let X, Y be vector fields on P such that their restrictions to M areequal to X,Y . Then ∇XY = (∇X Y )‖ and the second fundamental form h is

the vector-valued symmetric 2-tensor defined by ∇X Y = ∇XY + h(X,Y ), i.e.

h(X,Y ) := (∇X Y )⊥.The mean curvature vector H of M , is given by

Hα = gijhαij = ∆gFα + gijΓαβγ

∂F β

∂xi

∂F γ

∂xj,

where the Laplace Beltrami operator ∆g for functions f on M is given by

∆gf = divg(∇f) = gij∇i∇jf.

Here, divg(Y ) := ∇iY i denotes the divergence of a vector field X on M .

7

Definition 2.1. Let P be a Riemannian manifold, M a closed manifold, andF0 : M → P a smooth immersion. The mean curvature flow of F0 is a smoothmap F : M × [0, T∗) → P such that F (·, t) is an immersion for all t ∈ [0, T∗)solving the equation

∂

∂tF (p, t) = H(p, t), in M × (0, T∗), (1)

with initial condition F (·, 0) = F0.

This is a quasilinear, degenerated parabolic system. For P = Rn, equation(1) can be written as

∂

∂tF (p, t) = ∆gF (p, t), (2)

where the Laplacian ∆g is depending on pull-back metric g(·, t) induced byF (·, t). Hence we have a kind of heat equation and can expect that the equationdescribes some sort of diffusion process, which typically has a smootheningeffect.

2.2 First Variation Formula

The metric g on M , induces the measure dµ =√

det g dx1 ∧ · · · ∧ dxn, and thearea of the immersion F is defined by

Area(F ) :=

∫M

dµ.

Consider a smooth variation, i.e. a family of diffeomorphism φ(·, t) : M → Mfor t ∈ (−ε, ε) such that φ(·, 0) = id. The vector field Y := d/dt|t=0φ(·, t) iscalled the infinitesimal generator of the variation φ. The first variation of thearea functional then takes the form

∂

∂t

∣∣∣∣t=0

Area(F φ(·, t)) =

∫M

div(Y ‖)− g(H,Y ) dµ

If we additionally assume that the variation has compact support, i.e. thereexists a compact set K ⊂ M such that φ(p, t) = p for all p ∈ M \ K, thedivergence theorem implies

∂

∂t

∣∣∣∣t=0

Area(F φ(·, t)) = −∫M

g(H,Y ) dµ. (3)

By this formula, the mean curvature vector H is the gradient of the areafunctional with respect to the scalar product on L2(µ). Hence the mean cur-vature flow looks like the L2 gradient flow of the area functional. This is notstrictly speaking true, since the scalar product on L2(µt) depends on t. How-ever, it supports the physical intuition that the mean curvature flow reduces thearea in the most efficient way.

2.3 Tangential Motion

Clearly, the flow generated by equation (1) is a normal flow since the meancurvature is normal to M . An additional tangential component only changesthe immersions by a diffeomorphism of M . As we will see below, this can beused to achieve a strictly parabolic system.

8

Proposition 2.2. Let F : M × [0, T ) → P be a family of smooth immersionswhich satisfy

∂

∂tF (p, t) = H(p, t) +X(p, t), (p, t) ∈M × [0, T ),

F (p, 0) = F0(p), p ∈M,

(4)

where X is a smooth vector field tangent to F i.e. X(p, t) ∈ dF (p, t)(TpM).If M is compact, there exists a family of diffeomorphism φ(·, t) : M →M whichsmoothly depends on time t ∈ [0, T ), such that φ(·, t) = idM and F (p, t) =F (φ(p, t), t) satisfies the normal mean curvature equation (1).

If M is not compact, for every point (p, t) there exists a local diffeomorphismsuch that F (φ(p, t), t) satisfies the normal mean curvature equation.

Conversely, if F is a smooth mean curvature flow and φ(·, t) is a family ofdiffeomorphism which smoothly depends on time t ∈ [0, T ), then there exists atime dependent vector field X with X(p, t) ∈ dF (p, t)(TpM), such that the map

F (p, t) = F (φ−1(p, t), t) satisfies equation (4).

Proof. Since X(p, t) ∈ dF (p, t)(TpM) the vector field

Y (p, t) = −(dF (p, t))−1X(p, t)

is globally well defined and smooth. We consider the ordinary differential equa-tion on M

∂

∂tφ(p, t) = Y (p, t), (p, t) ∈M × [0, T ),

φ(·, 0) = id, p ∈M.(5)

If M is compact, there exists a unique smooth solution of this ODE system.Assume F solves equation (4), then F (p, t) = F (φ(p, t), t) satisfies

∂F

∂t(p, t) = dF (φ(p, t), t)Y (φ(p, t), t) + H(φ(p, t), t) +X(φ(p, t), t)

= H(φ(p, t), t)

= H(p, t).

2.4 Short Time Existence

The short time existence follows from Gage and Hamilton [26]. They provedthe following general result using a technique of Hamilton [31]

Theorem 2.3. Let M be a compact manifold and F0 : M → P be a smoothimmersion. Then in a positive time interval [0, t∗) there exists a unique smoothsolution of the heat equation

∂

∂tF = H(F (p, t)),

with initial condition F (·, 0) = F0.

9

Another way to prove the short time existence is given Huisken and Polden[37]. They consider a smooth immersion F0 : M → Rn+1 of a compact n-dimensional manifold M . In this case, given an orientation of M , there existsa unique normal ν. It turns out, that the solution can locally be written as agraph for a short time,

F (p, t) = F0(p) + f(p, t)ν(p),

and the height function f : M× [0, ε]→ R satisfies a strictly parabolic equation.A third way to prove short time existence is to use De Turck’s trick [13]. The

idea is to use an appropriate diffeomorphism φ to make the equation strictlyparabolic,

F (φ(p, t), t) = F (p, t).

If the diffeomorphism is generated by the vector field Y , i.e. it solves (5), thenequation (2) is equivalent to

∂

∂tF (p, t) = ∆gF (p, t)− Y k∇kF (p, t),

and with the choice Y k := gij(Γkij − Γ′kij) this system is strictly parabolic.

10

3 Curve Shortening Flow

In this chapter we study the smooth curve shortening flow and develop all of itsproperties which we will need later.

In the first section we induce the smooth flow, discover the evolution equationof basic quantities and state other well known properties of the flow. Thesecond section includes estimates of higher derivatives of the curvature, whichare proven by Altschuler [7] for the curve shortening flow of space curves andby Ecker and Huisken [20] for the mean curvature flow of hypersurfaces. As aconsequence of these estimates, the smooth flow exists as long as the curvaturestays bounded.

In section 3.3, we show the number of intersections of a curve shorteningflow with a hyperplane or a shrinking sphere is non increasing. Such intersectionnumbers were first used by Angenent in [6] for a general flow of curve in surfaces,where he proves an even stronger monotonicity. It is a major property that onlyis available for curves, and does not holds nor could be formulated for the meancurvature flow in general. Therefore, it distinguish the curve shortening flowfrom the mean curvature flow of higher dimensional manifolds.

In section 3.4, we analyze the flow at the first singular times. We willdiscover a modulus of continuation for the flow in time direction and prove thewell known fact that it converges uniformly to a Lipschitz curve at the firstsingular time.

In section 3.5, we follow Angenent [5] to obtain Lp estimates of the curvaturethat via a Moser iteration will lead to the following curvature bound

|k| ≤ Ct−1/2p,

where C depends on p > 1, on the time interval [0, t0] where this inequalityholds and on the Lp norm of the curvature at each time in [0, t0]. This estimatewill imply a short time existence result for curves in H2,p(S1,Rn). Note thatAngenent’s work [5, 6] is about a general flow of curves on surfaces. This generalflow includes the curve shortening flow. We always specialize his results to thecurve shortening flow in an Euclidean space and generalize them to arbitraryco-dimension.

In section 3.6, we study the curve shortening flow of a graph and state atheorem from Altschuler and Grayson [8], which implies a local curvature boundfor the graph flow with a small Lipschitz constant. Under a slope and curvaturebound, the evolution equation of the graph function is strictly parabolic. Thisleads by standard theory to a local C∞ estimate.

In section 3.7, we induce Lipschitz ε−graphs. This are curve such thatevery segment of length ε is the graph of a Lipschitz function. We show thatthis property is preserved under the flow for a short time. This is alreadyproven by Altschuler-Grayson [8] for space curves, by an argument that usesthe intersection number. Our proof uses directly the maximum principle andallows us to compute the time dependency of the Lipschitz constant. Togetherwith the local estimate from section 3.6, this will lead to a curvature bound forLipschitz ε−graphs with small Lipschitz constant of the form

|k| ≤ Ct−1/2,

on the time interval [0, ε2], where C is depending on nothing. A similar estimateis proven by Morgan and Tian [46] for the curve shortening flow in a compact

11

Riemannian manifold that evolves under the Ricci flow. The estimate can beseen as a improved version of the estimate of section 3.5 in the limit p to one, andanalogously it will implies a short time existence result for Lipschitz ε−graphswith small Lipschitz constant.

Section 3.8, includes a criterion when singularities can be formed and abound of the number of singular points that can arise at the first singular time.Both resolutes are proven by Angenent [5, 6] for the flow of curve in a surface.Again we specialize and generalize this into our setting. The result of thissection can be compared with section 5.6, where we analyze the weak solutionat a singular time. For general singular time, apart from a constant, the sameresult holds, but it turns out to be a little bit more involved since the singulartimes may have a accumulation point.

In section 3.9, we will prove a long time existence result for curves in Rnthat satisfy a convexity condition. Namely for curves such that the projectiononto a two-plane is a continuous embedded, uniformly convex curve. We showthat this property is preserved under the flow and that the limit at the firstsingular time is contained in a line. This extends long time existence resultsof the co-dimension one case, where convexity of the flow is naturally defined.This extends the following well known result of the co-dimension one case, whereconvexity of the flow is naturally defined. The mean curvature flow preservesthe convexity and for an initial convex hypersurface the smooth flow exists untilit shrinks to a round point. This is proven by Huisken in [34] for hypersurfacesand by Gage and Hamilton in [26] for curves in R2.

In section 3.10, we prove an ε−regularity theorem, which was conjecturedby Tom Ilmanen. This theorem implies a local curvature bound provided theL2 norm of the curvature in a parabolic cylinder is small enough. A relatedresult for the flow xt = xuu/|xu|2 is proven by Deckelnick in [12]. Other relatedresult can be fund in Ecker [17, 19], Ilmanen [39] and Han-Sun in [33].

3.1 Definition and Basic Properties

From now on, we consider the mean curvature flow of closed immersed curvesin arbitrary co-dimension. That is to say, we consider definition 2.1 for smoothimmersions F0 : S1 → Rn. In this case the mean curvature flow is called curveshortening flow. We change the notation and write the immersion as

x0 : S1 → Rn, u 7→ x0(u).

Moreover, by the one dimensionality we do not have to distinguish betweendifferent curvatures, and write k for the vector valued curvature of a curve.

Definition 3.1. Let I ⊂ R be an interval, which is either open, semi-open, orclosed, with non-empty interior. A smooth map x : S1 × I → Rn is called acurve shortening flow, if x(·, t) is an immersion for all t ∈ I and if it solves theequation

∂

∂tx(u, t) = k(u, t), (6)

in S1 × I.

By theorem 2.3, for every smooth immersed curve x0 there exists a closed setI = [0, t∗] and a curve shortening flow x : S1 × [0, t∗]→ Rn, which satisfies the

12

initial condition x(·, t) = x0. By extending this solution, there exists a maximaltime interval I = [0, T∗) where a smooth solution exists. Such a solution iscalled maximal, and we always denote the first singular time by T∗.

The arc-length parameter s(u, t) of an immersion x(·, t) : [0, 2π] → Rn isdefined by

s(u, t) :=

∫ u

0

|xu(v, t)|dv.

Sometimes we use the notation fu, fs and ft for the derivatives of a functionf(u, t) with respect to u, s and t. In this notation the tangent vector andcurvature is given by T := xs and k := xss, respectively. Hence the evolutionequation (6) can be written in the form of a heat equation

xt = xss.

Note that the arc-length parameter s depends on t.We introduce the Frenet-Serret frame (T,N,B1, ..., Bn−2) of a space curve,

where T , N , and Bi denotes the tangent-, normal-, and bi-normal unit vectors.If the curvature k = Ts =: |k|N does not vanishes, this defines the normal vectorN . A Frenet-Serret frame is an orthonormal frame satisfying the equation

∂

∂s

TNB1

...Bn−2

=

0 |k|−|k| 0 τ1

−τ1 0τn−2

−τn−2 0

TNB1

...Bn−2

. (7)

A Frenet-Serret frame is uniquely defined, if the curvature k and all torsions τifor i = 1, ..., n− 2 does not vanish.

Lemma 3.2. Under the curve shortening flow, there holds the interchange rule

∂

∂t

∂

∂s=

∂

∂s

∂

∂t+ |k|2 ∂

∂s, (8)

and the evolution equations

|xu|t = −|k|2|xu|, (9)

Tt = Tss + |k|2T, (10)

kt = kss + 2|k|2k + (|k|2)sT, (11)

(|k|2)t = (|k|2)ss − 2|ks|2 + 4|k|4, (12)

and if the curvature does not vanish, there hold

(|k|2)t = (|k|2)ss − 2(|k|s)2 − 2τ21 |k|2 + 2|k|4, (13)

|k|t = |k|ss + |k|3 − τ21 |k|. (14)

Proof. Note that by definition of the arc-length there holds ∂/∂s = |xu|−1∂/∂u.To prove (9) we compute

|xu|t =〈xu, xut〉|xu|

=〈xu, ku〉|xu|

= 〈T, ks〉 |xu|. (15)

13

Since |T |2 = |xs|2 = 1, there holds 〈T, k〉 = 0 and 〈T, ks〉 = −〈Ts, k〉 = −|k|2,which together with (15) implies (9). The interchange rule follows by (9) via

∂

∂t

∂

∂s=

∂

∂t

1

|xu|∂

∂u=

1

|xu|∂

∂t

∂

∂u− |xu|t|xu|2

∂

∂u

=1

|xu|∂

∂u

∂

∂t+|k|2

|xu|∂

∂u=

∂

∂s

∂

∂t+ |k|2 ∂

∂s

Equation (10) follows directly from the interchange rule

Tt = xst = xts + |k|2T = ks + |k|2T = Tss + |k|2T.

Equation (11) follows from (10)

kt = Tst = Tts + |k|2Ts = (Tss + |k|2T )s + |k|2k = kss + 2|k|2k + (|k|2)sT.

By equation (11) it follows

(|k|2)t = 2 〈k, kt〉 = 2 〈k, kss〉+ 4|k|4

= (|k|2)ss − 2|ks|2 + 4|k|4,

In the Frenet-Serret frame there holds ks = −|k|2T + |k|sN+τ1|k|B1, and hencewe can substitute |ks|2 = |k|4 + (|k|s)2 + τ2

1 |k|2 in equation (12). For the lastevolution equation we compute

|k|t = 〈k,N〉t = 〈kt, N〉 = 〈kss, N〉+ 2|k|3,|k|ss = 〈kss, N〉+ 2 〈ks, Ns〉+ 〈k,Nss〉 = 〈kss, N〉 − 〈k,Nss〉

= 〈kss, N〉 − 〈k, (−|k|T + τ1B1)s〉 = 〈kss, N〉 −⟨k,−|k|2N − τ2

1N⟩

= 〈kss, N〉+ |k|3 + τ21 |k|.

Clearly, the curve shortening flow shortens the length of a curve. To statethis accurately, we introduce the Radon measures µt on S1 by

µt(φ) :=

∫S1

φ(u)|xu(u, t)|du,

where φ ∈ C0(S1,R).

Lemma 3.3. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 0 ≤ t1 ≤ t2 < T∗ and u ∈ S1 there holds

|xu(u, t1)| ≥ |xu(u, t2)| ≥ e−C2(t2−t1)|xu(u, t1)|, (16)

where C := sup|k(u, t)| : t ∈ [t1, t2].In particular, for all 0 ≤ t1 ≤ t2 < T∗ there hold Lip(x(·, t2)) ≤ Lip(x(·, t1))

and µt2 ≤ µt1 in the sense of Radon measures.

Proof. Fix 0 ≤ t1 ≤ t2 < T∗ and u ∈ S1. By the evolution equation (9), thereholds

0 ≥ |xu(u, t)|t ≥ −C2|xu(u, t)|,

14

for all t ∈ [t1, t2]. The map t 7→ |xu(u, t)| is a non-increasing function for everyu ∈ S1. This shows the first inequality of (16). The second inequality followsby integration of f(t) := log(|xu(u, t)|), t ∈ [t1, t2]. Since f ′ ≥ −C2, there holds

f(t2)− f(t1) =

∫ t2

t1

f ′(t) dt ≥ −C2(t2 − t1),

and it follows

|xu(u, t2)| = exp(f(t2)) ≥ exp(f(t1)− C2(t2 − t1)) = e−C2(t2−t1)|xu(u, t1)|.

By the first inequality of (16), the Lipschitz constant

Lip(x(·, t)) := supu∈S1

|xu(u, t)|,

is non-increasing. By the monotonicity of the Lebesgue integral, there holds

µt2(φ) =

∫S1

φ(u)|xu(u, t2)|du ≤∫S1

φ(u)|xu(u, t1)|du = µt1(φ),

for every φ ∈ C∞(S1, [0,∞)).

The total absolute curvature K(t) of the flow, is defined by

K(t) :=

∫S1

|k(u, t)|dµt(u).

The following monotonicity of the total absolute curvature was proven byAltschuler [7], for curves in R3. This monotonicity will later be important tothe definition of weak solutions.

Lemma 3.4. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then the total absolute curvature K(t) is a non-increasing function.

Proof. For convenience, we first assume that k 6= 0, such that we can use equa-tion (14). Together with the evolution of the measure (9) this implies

d

dt

∫S1

|k| dµt =

∫S1

|k|t − |k|3 dµt ≤ −∫S1

τ21 |k| dµt.

If k does vanish at some point, we replace |k| by the quantity f :=√|k|2 + ε2

and compute by (12)

ft = fss −|ks|2(|k|2 + ε2)− 〈k, ks〉2

f3+

2|k|4

f.

Since 〈T, k〉 = 0 there holds |k|2 = −〈T, ks〉 and it follows

|k|2|ks|2 = |k|2(|k⊥s |2 + 〈T, ks〉2) ≥ 〈k, ks〉2 + |k|2〈T, ks〉2 = 〈k, ks〉2 + |k|6.

This implies the estimate

ft ≤ fss −ε2|ks|2 + |k|6

f3+

2|k|4

f.

15

Integrating this leads to

d

dt

∫S1

f dµt =

∫S1

ft − |k|2f dµt

≤∫S1

−ε2|ks|2 − |k|6 + 2|k|4f2 − |k|2f4

f3dµt

=

∫S1

−ε2|ks|2 − (f2 − |k|2)2|k|2

f3dµt

=

∫S1

−ε2|ks|2 − ε4|k|2

f3dµt ≤ 0,

and since this holds for every ε > 0, this implies the lemma.

3.2 Estimates for Higher Derivatives

In this section we use the notation Du, Dt and Ds for the derivatives. Next weimprove the interchange rule from lemma 3.2.

Lemma 3.5. Under the curve shortening flow, there holds the interchange rule

DtDls = Dl

sDt +

l−1∑i=0

(l

i+ 1

)(Di

s|k|2)Dl−is (17)

for all l ≥ 1. Moreover, for general m, l ≥ 1 there holds

Dmt D

ls = Dl

sDmt +

m−1∑j=0

l−1∑i=0

Dm−1−jt Di

s|k|2Dl−is Dj

t . (18)

Proof. We first prove that (18) holds for m = 1, i.e. we claim that

DtDls = Dl

sDt +

l−1∑m=0

Dms |k|2Dl−m

s , (19)

holds for all l ≥ 1.Proof by induction. For l = 1, equation (19) follows by lemma 3.2. Assuming

(19) holds for l, we compute

DtDl+1s =

(DlsDt +

l−1∑m=0

Dms |k|2Dl−m

s

)Ds

= Dls(DsDt + |k|2Ds) +

l−1∑m=0

Dms |k|2Dl+1−m

s

= Dl+1s Dt +Dl

s|k|2Ds +

l−1∑m=0

Dms |k|2Dl+1−m

s

= Dl+1s Dt +

l∑m=0

Dms |k|2Dl+1−m

s .

16

Equation (17) follows by (19) via

l−1∑m=0

Dms |k|2Dl−m

s =

l−1∑m=0

m∑i=0

(m

i

)(Di

s|k|2)Dl−is

=

l−1∑i=0

(l−1∑m=i

(m

i

))(Di

s|k|2)Dl−is

=

l−1∑i=0

(l−i∑m=1

(m+ i− 1

i

))(Di

s|k|2)Dl−is

=

l−1∑i=0

(l

i+ 1

)(Di

s|k|2)Dl−is .

In the last step we have used

n∑m=1

(m+ i− 1

i

)=

(n+ i

i+ 1

),

which also follows by induction via

n+1∑m=1

(m+ i− 1

i

)=

(n+ i

i+ 1

)+

(n+ i

i

)=

(n+ i+ 1

i+ 1

).

Equality (18) follows again by induction. We have already proven that (18)holds for m = 1 and all l ≥ 1. Assuming (18) holds for m, l ≥ 1, we compute

Dm+1t Dl

s = Dt

DlsD

mt +

m−1∑j=0

l−1∑i=0

Dm−1−jt Di

s|k|2Dl−is Dj

t

=

(DlsDt +

l−1∑m=0

Dms |k|2Dl−m

s

)Dmt +

m−1∑j=0

l−1∑i=0

Dm−jt Di

s|k|2Dl−is Dj

t

= DlsD

m+1t +

l−1∑m=0

Dms |k|2Dl−m

s Dmt +

m−1∑j=0

l−1∑i=0

Dm−jt Di

s|k|2Dl−is Dj

t

= DlsD

m+1t +

m∑j=0

l−1∑i=0

Dm−jt Di

s|k|2Dl−is Dj

t .

The following dilation invariant estimate for higher derivatives of the curva-ture is from Altschuler [7].

Theorem 3.6. (Altschuler [7], Theorem 3.1) Let x : S1 × [0, T∗) → Rn be asmooth curve shortening flow and let Mt := sup |k(·, t)|2. Then for all l ∈ N0

there exists a constant Cl, depending only on l, such that for every t0 ∈ [0, T∗)there holds ∣∣Dl

sk(u, t)∣∣2 ≤ ClMt0

(t− t0)l, (20)

for all (u, t) ∈ S1 × [t0, t0 + 1/(8Mt0)].

17

Proof. By a translation x(u, t) := x(u, t0 + t) we can assume that t0 = 0. Wefirst consider the case l = 0. By the evolution equation (12) we have

Dt|k|2 = D2s |k|2 − 2|Dsk|2 + 4|k|4.

Since |k|2 = −〈T,Dsk〉 there holds

Dt|k|2 ≤ D2s |k|2 + 2|k|4.

Hence, by the maximum principle it follows that

DtMt ≤ 2M2t ,

and integration leads to the estimate

Mt ≤Mt0

1− 2tMt0

, (21)

for all t ∈ [0, 1/(2M0)]. In particular, for t ∈ [0, 1/(4M0)] we have

Mt ≤ 2M0.

In the case l = 1, we observe that the quantity f := t|ks|2 + 4|k|2 satisfies

∂

∂tf ≤ ∂2

∂s2f + 64|k|4,

on S1 × [0, 1/(8M0)]. Hence by the maximum principle there holds

t|ks|2 + 4|k|2 ≤ 4M0 + 64M20 t ≤ 12M0.

For l ≥ 2 we reduce the problem by scaling x(u, t) := λx(u, λ−2t) withλ :=

√M0. Hence we can assume that M0 = 1.

We argue by induction. Fix m ≥ 1 and assume (20) already holds forl = 0, ...,m − 1. We aim to show that there exists a constant Cm, dependingonly on m, such that there holds tm|Dm

s k| ≤ Cm on S1 × [0, 1/8].By lemma 3.5 we compute

Dt|DlsT |2 = 2

⟨DtD

l+1s x,Dl

sT⟩

= 2⟨Dl+1s Dtx,D

lsT⟩

+ 2

l∑i=0

(l + 1

i+ 1

)(Di

s|k|2)⟨Dl+1−is x,Dl

sT⟩

= 2⟨Dl+3s T,Dl

sT⟩

+ 2

l∑i=0

(l + 1

i+ 1

)(Di

s|DsT |2)⟨Dl−is T,Dl

sT⟩

= D2s |Dl

sT |2 − 2|Dl+1s T |2 + 2

l∑i=0

(l + 1

i+ 1

)(Di

s|k|2)⟨Dl−is T,Dl

sT⟩.

This implies(Dt −D2

s

)(tl|Dl

sk|2) = ltl−1|Dlsk|2 + tl

(Dt −D2

s

)(|Dl+1

s T |2)

= ltl−1|Dlsk|2 − 2tl|Dl+1

s k|2 (22)

+ 2

l+1∑i=0

(l + 2

i+ 1

)tl(Di

s|k|2)⟨Dl+1−is T,Dl

sk⟩.

18

By induction hypotheses there holds ti|Disk|2 ≤ Ci on S1 × [0, 1/8] for every

i = 0, ...,m− 1. Assuming that l = 0, ...,m, we estimate the last term by

l+1∑i=0

i∑j=0

2

(l + 2

i+ 1

)(i

j

)tl|Dj

sk| |Di−js k| |Dl+1−i

s T | |Dlsk|

=

l+1∑j=0

2

(l + 1

j

)tl|Dj

sk||Dl+1−js k| |Dl

sk|

+

l∑i=0

i∑j=0

2

(l + 2

i+ 1

)(i

j

)tl|Dj

sk| |Di−js k| |Dl−i

s k| |Dlsk|

= 4tl|k| |Dl+1s k| |Dl

sk|+ 4(l + 1)tl|Dsk| |Dlsk|2

+

l−1∑j=2

2

(l + 1

j

)tl|Dj

sk||Dl+1−js k| |Dl

sk|

+ 6(l + 2)tl|k|2 |Dlsk|2 +

l−1∑j=2

2(l + 2)

(l

j

)tl|k| |Dj

sk| |Dl−js k| |Dl

sk|

+

l−1∑i=1

i∑j=0

2

(l + 2

i+ 1

)(i

j

)tl|Dj

sk| |Di−js k| |Dl−i

s k| |Dlsk|

≤ tl|Dl+1s k|2 + 4C0t

l|Dlsk|2 + 4(l + 1)C1t

l−1|Dlsk|2

+

l−1∑j=2

(l + 1

j

)(CjCl+1−j + tl−1|Dl

sk|2)

+ 6(l + 2)C0tl|Dl

sk|2 +

l−1∑j=2

2(l + 2)

(l

j

)(C0CjCl−j + tl|Dl

sk|)

+

l−1∑i=1

i∑j=0

(l + 2

i+ 1

)(i

j

)(CjCi−jCl+1−i + tl|Dl

sk|2).

Together with (22) this implies that there exists a constant Al, depending onlyon l and C0, C1, and a constant Bl, depending only on l and C0, ..., Cl−1 suchthat

(Dt −D2s)(t

l|Dlsk|2) ≤ −tl|Dl+1

s k|2 +Altl−1|Dl

sk|2 +Bl,

holds on S1 × [0, 1/4]. Since we already have proven that (22) holds for C0 = 2and C1 = 12, the constant Al depends only on l. By induction assumptionCi, i = 0, ..,m − 1 only depends on i, hence also Bl depends only on l, for alll = 0, ..,m.

The function

fm :=

m∑l=0

al;mtl|Dl

sk|2

19

where al;m is define inductively by am;m := 1 and ai−1;m := ai;mAi/8, satisfies

(Dt −D2s)f ≤

m∑l=0

(al;mAlt− al−1;m) tl−1|Dlsk|2 +

m∑l=0

al;mBl

≤m∑l=0

al;mBl =: Gm,

on S1 × [0, 1/8]. By the maximum principle there holds

supS1

tl−1|DlsT (·, t)|2 ≤ sup

S1

f(·, t) ≤ supS1

f(·, 0) +Gmt ≤ a0;m +Gm/8 =: Cm,

for all t ∈ [0, 1/8].

Now we show that a bound on the arc-length derivative of the curvatureimplies a bound of all derivatives of the parametrization x(u, t).

Lemma 3.7. Let x : S1 × [0, t∗] → Rn be a smooth curve shortening flow andassume there exists t0 ∈ [0, t∗] such that x(·, t0) is parametrized by arc-length,i.e. |xu(·, t0)| = 1. Let I ⊂ S1 be a compact interval and assume there existconstants Cl, l ≥ 0 such that∣∣Dl

sk(u, t)∣∣ ≤ Cl, (u, t) ∈ I × [0, t∗]. (23)

Then for all (m, l) ∈ N20 \ 0 there exists a constant Km;l, depending only on

(m, l), t∗ and Ci for i = 0, ..., 2m+ l − 1 such that∣∣Dmt D

lux(u, t)

∣∣ ≤ Km;l, (u, t) ∈ I × [0, t∗]. (24)

Remark. Sometimes we use lemma 3.7 in the case x(·, t0) is parametrized byconstant speed, i.e. |xu(·, t0)| = c. In this case, by a re-parametrization, itimmediately follows∣∣Dm

t Dlux(u, t)

∣∣ ≤ c−lKm;l, (u, t) ∈ I × [0, t∗].

Proof.

A) Claim. There exists a constant Λ0 depending only on t∗ and C0 such that

|xu| ≤ Λ0, S1 × [0, t∗].

Proof. By lemma (3.3) there holds

e−C20 (t2−t1)|xu(u, t1)| ≤ |xu(u, t2)| ≤ |xu(u, t1)|,

for all t1 ≤ t2 and u ∈ S1. Since x(·, t0) is parametrized by arc-length thisimplies

|xu(u, t)| ≤ Λ0 := eC20 t∗ , (u, t) ∈ I × [0, t∗].

B) Claim. For all l ≥ 1 there exists a constant Λl depending only on l, t∗ andCi for i = 0, ..., l such that∣∣Dl

s|xu|∣∣ ≤ Λl, I × [0, t∗]. (25)

20

Proof. We simplify the notation by a(u, t) := |xu(u, t)|. We first considerthe the case l = 1. By (9) and (8) there holds

DtDsa = DsDta+ |k|2Dsa = Ds(−|k|2a) + |k|2Dsa = −(Ds|k|2)a.

Since x(·, t0) is parametrized by arc-length there holds Dsa(u, t0) = 0, andintegration leads to

|Dsa(u, t)| ≤∣∣∣∣∫ t

t0

DtDsa(u, t′)dt′∣∣∣∣ ≤ Λ1 := 2Λ0C0C1t∗,

for all (u, t) ∈ I × [0, t∗].

For l ≥ 2, we argue by induction. Assume (25) holds already for 1, .., l − 1.By lemma 3.5 we compute

DtDlsa = Dl

sDta+

l−1∑i=0

(l

i+ 1

)(Di

s|k|2)Dl−is a

= Dls(−|k|2a) +

l−1∑i=0

(l

i+ 1

)(Di

s|k|2)Dl−is a

= −l∑i=0

(l

i

)(Di

s|k|2)Dl−is a+

l−1∑i=0

(l

i+ 1

)(Di

s|k|2)Dl−is a

=

l∑i=0

l − 2i− 1

i+ 1

(l

i

)(Di

s|k|2)Dl−is a

= (l − 1)|k|2Dlsa+

l∑i=1

i∑j=0

l − 2i− 1

i+ 1

(l

i

)(i

j

)⟨Djsk,D

i−js k

⟩Dl−is a.

This proves the estimate∣∣Dt|Dlsa|∣∣ ≤ |DtD

lsa| ≤ (l − 1)C2

0 |Dlsa|+A,

on I × [0, t∗], where

A :=

l∑i=1

i∑j=0

(l

i

)(i

j

)|l − 2i− 1|i+ 1

CjCi−jΛl−i.

Note that by the induction assumption A only depends on l, t∗ and Ci fori = 0, ..., l. Since Dl

sa(u, t0) = 0, by Gronwall’s inequality this implies

|Dlsa| ≤ Λl := At∗ exp

((l − 1)C2

0 t∗),

on I × [0, t∗].

C) Claim. For all m, l ≥ 0 there exist a constant Cm;l depending only on (m, l)and Ci for i = 0, ..., 2m+ l such that

|Dmt D

lsk| ≤ Cm;l, I × [0, t∗] (26)

21

Proof. Note that for m = 0 the claim holds by assumption with C0,l = Cl.Let m > 0 and assume the claim holds for 0, ..,m − 1 and all l ≥ 0. Againby lemma 3.5 there holds

Dmt D

lsk = Dm

t Dl+2s x

= Dm−1t Dl+2

s Dtx+

l+1∑i=0

(l + 2

i+ 1

)Dm−1t

((Di

s|k|2)Dl+2−is x

)= Dm−1

t Dl+2s k +

l∑i=0

(l + 2

i+ 1

)Dm−1t

((Di

s|k|2)Dl−is k

)+ (l + 2)Dm−1

t

((Dl+1

s |k|2)T)

The right hand side depends only on DitD

jsk for i = 0, ...,m − 1 and j =

0, ..., l + 2, and hence by assumption is bounded by a constant that onlydepends on Ci for i = 0, ..., 2(m− 1) + (l + 2).

D) Claim. For all m ≥ 1 and l ≥ 0 there exists a constant Λm;l depending onlyon (m, l), t∗ and Ci for i = 0, ..., 2(m− 1) + l such that there holds∣∣Dm

t Dls|xu|

∣∣ ≤ Λm;l, I × [0, t∗]. (27)

Proof. We again use the shortcut a = |xu|. By lemma 3.5 there holds

Dmt D

lsa = Dm−1

t DlsDta+

l−1∑i=0

(l

i+ 1

)Dm−1t

((Di

s|k|2)Dl−is a

)= −Dm−1

t Dls(|k|2a) +

l−1∑i=0

(l

i+ 1

)Dm−1t

((Di

s|k|2)Dl−is a

)The right hand side depends only on Di

tDjsk and Di

tDjsa for i = 0, ...,m− 1,

j = 0, ..., l.

For m = 1, by claim B) and C), this implies that |DtDlsa| is bounded by

a constant that only depends on t∗ and Ci for i = 0, ..., l. For m ≥ 2, thistogether with claim C) implies the induction step. If 27 already holds for1, ..., ,m−1 and all l ≥ 0, then |Dm

t Dlsa| is bounded by a constant that only

depends on (m, l), t∗ and Ci for i = 0, ..., 2(m− 1) + l.

E) Claim. For all (m, l) ∈ N20 \ 0 there exist a constant Km;l depending only

on (m, l), t∗ and Ci for i = 0, ..., 2m+ l − 1 such that

|Dmt D

lux| ≤ Km;l, I × [0, t∗]. (28)

Proof. Note that by definition of the arc-length there holds Du = aDs. Wefirst consider the case m ≥ 1. In this case there holds

Dmt D

lux = Dm−1

t Dluk = Dm−1

t (aDs)lk.

The right hand side depends only on DitD

j+1s k and Di

tDjsa for i = 0, ...,m−1

and j = 0, ..., l−1. By claim C) and D), |Dmt D

lux| is bounded by a constant

that only depends on (m, l), t∗ and Ci for i = 0, ..., 2(m− 1) + l.

22

The special case m = 0 and l = 1 follows directly by claim A). For the casem = 0 and l ≥ 2 we compute

D2ux = (aDs)

2x = aDsaT = a2k + a(Dsa)T.

Therefore, we have

Dlux = (aDs)

l−2(a2k + a(Dsa)T )

The right hand side depends only on T , Disk for i = 0, .., l − 2 and Di

sa fori = 0, .., l− 1. Therefore, by claim B), |Dl

ux| is bounded by a constant thatonly depends on l, t∗ and Ci for i = 0, ..., l − 1.

As a consequence of the estimates above, the smooth flow can be extendedas long as the curvature stays bounded. The following corollary is proven byAltschuler and Grayson [8]. Note that they state it only for n = 3, but the prooffor general n ≥ 3 is identical. We reproduce their proof for convenience of thereader.

Corollary 3.8. (Altschuler-Grayson [8], Theorem 1.13) Let x : S1 × [0, t∗) →Rn be a smooth curve shortening flow. If the curvature is uniformly bounded,supS1×[0,t∗) |k| ≤ C0, then there exists an ε > 0 such that the smooth solution

can be extended to x : S1 × [0, t∗ + ε)→ Rn.

Proof. Let t0 := t∗ − 1/(12C20 ). Since t∗ < t0 + 1/(8C2

0 ), by theorem 3.6 thereexist constants Cl, l ≥ 1 depending only on l such that

|Dlsk| ≤ ClC0(t− t0)−l/2, S1 × [t0, t∗).

In particular, this implies

|Dlsk| ≤ Cl := (4

√3)lClC

l+10 , S1 × [t1, t∗),

where t1 := t∗−1/(16C20 ). By a chart we write x(·, t1) : [0, 2π]→ Rn and define

the map φ : [0, 2π]→ [0, L]

φ(u) :=

∫ u

0

|xu(v, t1)|dv,

where L := L(x(·, t1)). Since x(·, t1) is a smooth immersion, φ is diffeomorphism.Hence we can define

x(v, t) := x(φ−1(v), t), (v, t) ∈ [0, L]× [t1, t∗].

Since there holds |xu(u, t1)| = 1, by lemma 3.7 there exist constants Km;l de-

pending only on (m, l), t∗ − t1 and Ci for i = 0, ..., 2m+ l − 1 such that

|Dmt D

lux| ≤ Km;l, S1 × [t1, t∗).

Note that t∗ − t1 depends by definition only on C0, and Ci depends only on C0

and i. Hence Km;l effectively depends only on (m, l) and C0. This shows thatx(u, t) = x(φ(u), t) satisfies

|Dmt D

lux| ≤ Km;l, S1 × [t1, t∗),

23

for constants Km;l that only depend on (m, l), C0 and the Cl norm of x(·, t1).By integration of DtD

lux this shows that the maps x(·, t) converges to some

smooth curve x∗ ∈ C∞(S1,Rn) as t→ t∗.By lemma (9) there holds

|xu(u, t)| ≥ |xu(u, 0)|e−C20 t.

Hence x∗ is a smooth immersion and the corollary follows by the short timeexistence theorem 2.3.

Note that there are no bounds for the torsions are necessary to extend theflow. In fact the torsions τi are only bounded as long as the curvature is positive.

Lemma 3.9. Let τi be given by equation (7). Then for all l = 1, ..., n− 2 thereholds

|k|2l∏i=1

τ2i ≤

∣∣∣∣∂lk∂sl∣∣∣∣2 .

Proof. For l = 1, we compute in the Frenet-Serret frame

ks = −|k|2T + |k|sN + |k|τ1B1,

and hence |k|2τ21 ≤ |ks|2. For l > 1, since (Bl−1)s = −τl−2Bl−2 + τlBl it follows

by induction that ⟨∂lsk,Bl

⟩= |k|

l∏i=1

τi.

3.3 Intersection Theorem

In [6] Angenent proves, that the number of self-intersections of a curve on asurface is a monotone non-increasing quantity for general parabolic flows. Thefollowing theorem states this in the case of the curve shortening flow in theplane.

Theorem 3.10. Let x : S1 × [0, t∗] → R2 be a smooth curve shortening flow,and assume that 2π is the smallest period of x(·, 0). Then the number of self-intersections

i(t) := #(u1, u2) ∈ S1 × S1 : x(u1, t) = x(u2, t), u1 6= u2,

is finite for t > 0 and a non-increasing function of t ∈ (0, t∗]. It decreaseswhenever there exists a tangency. 1

For curves in higher co-dimension there is no analogon with self-intersections,but we still have the following well known theorem:

Theorem 3.11. Let x : S1 × [0, t∗] → Rn be a smooth curve shortening flowand let M(t) be either a constant hyperplane M(t) = P or a shrinking sphere

1i.e. the intersection is non transversal, see definition C.1.

24

M(t) = ∂Br(t)(p) with radius r(t) =√r20 − 2t. Then either x(S1, t) ⊂M(t) for

all t ∈ [0, t∗] or the number of intersections

i(t) := #u ∈ S1 : x(u, t) ∈M(t),

is finite for t ∈ (0, t∗] and a non-increasing function of t ∈ (0, t∗]. Moreover,if V = [u1, u2] is an interval such that x(u1, t), x(u2, t) /∈ M(t) for all timet ∈ [0, t∗] then the number of intersection points in V

iV (t) := #u ∈ V : x(u, t) ∈M(t),

is finite for t > 0 and a non-increasing function of t ∈ (0, t∗].Moreover, the number of intersections decreases at t0 if there exists a tan-

gency between x(·, t0) and M(t).

Proof. By the short time existence, we can extend the smooth flow to x : S1 ×[0, t∗ + ε]→ Rn.

In the case of a hyperplane P := y ∈ Rn : 〈y, v〉 = c, we define thefunction f(u, t) := 〈x(u, t), v〉− c. This function satisfies the evolution equation

ft = 〈k, v〉 = fss,

where s denotes the arc-length of x(·, t).In the case of a shrinking sphere M(t) = ∂Br(t)(p), we consider the function

f(u, t) := |x− p|2 − r(t)2, and calculate

ft = 2 〈k, x− p〉+ 2 = fss.

Hence in both cases the function f satisfies the same evolution equation

ft = fss =1

|xu(u, t)|2fuu −

〈xuu(u, t), xu(u, t)〉|xu(u, t)|4

fu.

Using a diffeomorphism φ : V → [0, 1] and since by assumption x(·, t) is animmersion for every t ∈ [0, t∗ + ε], we can apply the following theorem provenin Angenent [4].

Theorem 3.12. Let f : [0, 1]× [0, T )→ R be a continuous classical solution ofthe equation

ft = a(u, t)fuu + b(u, t)fu + c(u, t)f,

with either periodic boundary conditions or f(0, t) 6= 0 and f(1, t) 6= 0 for allt ∈ [0, T ). In the case of periodic boundary conditions, assume that f(·, 0) 6= 0.Assume that the coefficients satisfy δ ≤ a ≤ δ−1 for some δ > 0 and

a, at, au, auu, b, bt, bu, c ∈ L∞([0, 1]× (0, T )).

Then the number of zero’s of f(·, t)

z(t) := #u ∈ [0, 1) : f(u, t) = 0

is finite for t ∈ (0, T ) and a non-increasing function of t ∈ (0, T ). In addition,if for some t0 ∈ (0, T ), f(·, t0) has a multiple zero, i.e. if there exists u0 ∈ [0, 1]such that f(u0, t0) = 0 and fu(u0, t0) = 0, then there holds z(t1) > z(t2) for allt1 < t0 < t2.

25

Remark. Note that by the implicit function theorem z(t) is locally constant ata time t0 where f(·, t0) does not have a multiple zero. Hence z(t) exactly dropsat t0 if and only if f(·, t0) have a multiple zero.

Clearly there holds f(u, t) = 0 if and only if x(u, t) ∈ M(t). If x(S1, 0) ⊂M(0) it follows f(·, 0) = 0 and by the maximum principle there holds f(·, t) = 0for all t ∈ [0, t∗]. Otherwise there holds f(·, 0) 6= 0 and theorem 3.12 withperiodic boundary conditions implies that iV (t) = z(t) is finite for t > 0 anda non-increasing function of t ∈ (0, t∗ + ε). In fact, a multiple zero of f(·, t0)corresponds to a tangential intersection point of x(·, t0) with M(t0). HenceiV (t) decreases whenever there exists a tangency. This proves the first part ofthe theorem.

To prove the second part of the theorem, we choose ε > 0 small enough suchthat there holds x(u1, t), x(u2, t) /∈M(t) for all time t ∈ [0, t∗ + ε]. Hence thereholds f(0, t) 6= 0 and f(1, t) 6= 0 for all t ∈ [0, t∗ + ε] and we can use theorem3.12 in this case and argue analogously.

The proof of the theorem 3.11 would actually work for the flow of totallyumbilic, complete and embedded hypersurfaces M(t). For fun, we show thatthese are exactly the spheres and hyperplanes.

Lemma 3.13. Let M → Rn+1, n ≥ 2 be a connected, totally umbilic, completeand embedded hypersurface, then either M is a sphere or a hyperplane.

Proof. We use the same notation as in the lemma above. Since M is totallyumbilic there holds nhij = Hgij at each point and by Codazzi it follows

∇iH = gjk∇k(Hgij) = gjk∇k(nhij) = ngjk∇ihjk = n∇iH.

Since n ≥ 2, the mean curvature H is constant on M . Now, let r be therestriction of the radius vector on M and ν be a unit normal ν on M . Then

∇iν = −hjiej = −(H/n)gji ej = −(H/n)∇ir.

Hence X := (H/n)r + ν is constant on M . If the mean curvature is not zero,this implies

|nX/H − r|2 = (n/H)2,

which defines a sphere around nX/H with radius |n/H|. Otherwise the unitnormal ν is constant on M and

∇i 〈ν, r〉 = 〈ν,∇ir〉 = 0,

which defines a hyperplane. Since M is connected, this shows that M is con-tained in a sphere or a hyperplane, and by completeness, it has to be a sphereor a hyperplane.

Theorem 3.11 has many applications, but it has the disadvantage that it isnot stable under a limit process. An advantage of the number of intersections isthat it allows us to argue very geometrically, where the differential equations andthe maximum principle are hidden. A first simple consequence is that a compactflow must blow up in finite time, for if x(S1, 0) ⊂ BR(p) then T∗ ≤ R2/2. Asecond consequence is the following lemma, which shows that the flow stays ina neighborhood for a short time interval. We define the neighborhood of a setA ⊂ Rn by

Nd(A) := p ∈ Rn : distRn(p,A) < d.

26

Lemma 3.14. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 0 ≤ t0 ≤ t < T∗ there holds

x(S1, t) ⊂ Nd(x(S1, t0)),

where d =√

2(t− t0). Moreover, if V = (a, b) is an open interval and B1, B2 ⊂Rn are closed sets which contain the boundary points x(a, t) ∈ B1 and x(b, t) ∈B2 for all time t ∈ [0, t0] then there holds

x(V, t) ⊂ Nd(x(V, t0) ∪B1 ∪B2).

Proof. We assume that t0 = 0. For each point p ∈ Rn with positive initialdistance d := distRn(p, x(S1, t0)) > 0, the sphere ∂Br(p) with radius r < ddoes not intersect the curve x(S1, t0). By theorem 3.11 the shrinking spherewith radius r(t) =

√r2 − 2t does not intersect with the curve x(S1, t). Hence

we have d(t) := distRn(p, x(S1, t)) ≥√r2 − 2t, and since this holds for all

r < d, we get d(t) ≥√d2 − 2t. Therefore, the flow does not reach the point

p before 2t = d2. For the local version, we again take p ∈ Rn with d :=distRn(p, x(V, t0) ∪ B1 ∪ B2) > 0. By assumption for t ∈ [0, t0] the boundarypoints are outside the ball Br(p) with radius r < d. In particular, the boundarypoints are disjoint from the shrinking sphere M(t) := ∂Br(t)(p) and local versionin theorem 3.11 implies that M(t) does not intersect with the curve x(V, t).Hence the result follows as above.

Since we work with immersed curves, we always count the number of pre-images of intersections

#x−1(M) := #u ∈ S1 : x(u) ∈M,

where M ⊂ Rn is an embedded manifold and either is complete or satisfiesx(S1) ∩ ∂M = ∅.

As a third application, we show that the number of intersections can be usedto get a curvature bound in a geometric way.

Lemma 3.15. Let x ∈ C2(S1,Rn) and 2R = diameter(x(S1)). If there existsan r < R such that

#x−1(∂Br(p)) ≤ 2, (29)

holds for all p ∈ Rn then the curvature of x is bounded by 1/r.

Remark. For an embedded curve x with bounded curvature, there exists a kindof reverse statement. In this case, there exists a ρ > 0, such that the tube withradius ρ around the curve has no self-intersections. For r < 1/ sup |k| and r < ρ,there holds #x−1(∂Br(p)) ≤ 2.

Proof. We assume by contradiction that there exists a point p = x(u) with|k(u)| > 1/r. Let N be the unit normal at p such that k(u) = |k(u)|N . We claimthat there exists λ ∈ (r,R) such that #x−1(∂Br(q)) ≥ 4, where q := p+ λN .

For λ > r the point p is not contained in the ball Br(q). Since |k(u)| > 1/rfor λ near r, the connected component of x(S1) \ Br(x)) which contains p iscontained in the closed ball Bλ(q).

Since r < R, we can assume that λ < R. Hence there is at least onecomponent of x(S1)\Bλ(q), thus x(S1)\Br(q)) has at least two components.

27

Corollary 3.16. Let x : S1 × [0, T∗)→ Rn be a smooth curve shortening flow.Assume there exists a r0 > 0 such that #x(·, 0)−1(∂Br0(p)) ≤ 2 for all p ∈ Rn.Then for all times t with r(t) :=

√r20 − 2t < diameter(x(S1, t))/2, there holds

supS1

|k(·, t)| ≤ 1/r(t).

The corollary follows directly from theorem 3.11 and the lemma above. Notethat the corollary is just given to illustrate the geometric argument. Actually, inthis case, it is better to use the evolution equation of |k|2. In fact, by equation(21) there holds

supS1

|k(·, t)| ≤(C−2

0 − 2t)−1/2

,

for all t ∈ [0, 1/(2C20 )], where C0 := supS1 |k(·, 0)|.

3.4 Convergence at the First Singular Time

The Hausdorff distance of two sets A,B is defined by

dH(A,B) := infr ≥ 0 : A ⊂ Nr(B), B ⊂ Nr(A).

Lemma 3.17. Let x : S1× [0, T∗)→ Rn be a maximal smooth curve shorteningflow. Then the image γ(t) := x(S1, t) converges in Hausdorff distance to theimage γ∗ of a rectifiable curve as t→ T∗.

Proof. Let ti T∗ be an increasing sequence and define xi(u) := x(u, ti). Bylemma 3.3 the sequence xi is uniformly Lipschitz and hence by Arzela Ascolithere exists a subsequence which converges in C0 to a Lipschitz curve x∗ andwe define γ∗ := x(S1).

For every fixed r > 0 there exists i0 ∈ N such that√

2(T∗ − ti0) ≤ r/2 andεi := ‖xi − x∗‖C0 ≤ r/2 for all i ≥ i0.

We claim that there holds dH(γ(t), γ∗) ≤ r for all t ∈ [ti0 , T∗]. By lemma3.14, there holds γ(t) ⊂ Nd(γ(ti0)), where d =

√2(t− ti0). Hence for all

t ∈ [ti0 , T∗) there holds

γ(t) ⊂ Nεi+√

2(t−ti0 )(γ∗) ⊂ Nr(γ∗).

On the other hand, for ever t ∈ [ti0 , T∗) there exists j ≥ i0 such that tj ∈ [t, T∗).

Again by lemma 3.14, there holds γ(tj) ⊂ Nd(γ(t)), where d =√

2(tj − t) andthis implies

γ∗ ⊂ Nεi+√

2(tj−t)(γ(t)) ⊂ Nr(γ(t)).

Lemma 3.18. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 0 ≤ t1 ≤ t2 < T∗ and every interval I = [u1, u2] ⊂ S1 withµt2(I) > 0,, there holds

|x(u1, t1)− x(u2, t2)| ≤ µt1(I) +

√µt1(I)|t1 − t2|

µt2(I).

28

and the second term is bounded by√µt1(I)|t1 − t2|

µt2(I)=

√µt1(I)|t1 − t2|

µt1(A)1/3|t1 − t2|1/3

≤ µt1(A)1/3|t1 − t2|1/3.

Together with the maximum principle the lemma implies the following mod-ulus of continuation.

Corollary 3.20. Let x : S1 × [0, T∗)→ Rn be a smooth curve shortening flow.Then for every u ∈ S1 and 0 ≤ t1 ≤ t2 < T∗ there holds

|x(u, t1)− x(u, t2)| ≤ 2

∫ t2

t1

∫S1

|k|2 dµtdt+ 3µt1(S1)1/3|t1 − t2|1/3.

Proof. We simplify the notation L(t) := µt(S1) and

m(σ) :=

∫ t1+σ

t1

∫S1

|k|2 dµtdt.

Moreover, we define σ2 := t2 − t1, σ3 := supσ ∈ [0, T∗ − t1) : m(σ) < L(t1)/2and σ4 := minL(t1)2/8, σ3.

First we consider the case σ2 ≤ σ4. By definition of σ3 and σ4 there holds

L(t1)− L(t2) = m(σ2) ≤ L(t1)/2.

Therefore, there holds L(t2) ≥ L(t1)/2 and it follows

t2 − t1 = σ2 ≤ σ4 ≤ L(t1)2/8 ≤ L(t2)3/L(t1).

Hence we can use lemma 3.19 to conclude

|x(u, t1)− x(u, t2)| ≤ 2m(σ2) + 3L(t1)1/3σ1/32 ,

which proves the corollary in this case.For the second case σ2 ≥ σ4, we use the maximum principle. Since x(S1, t1)

is contained in a ball with diameter L(t1)/2 there holds

|x(u, t1)− x(u, t2)| ≤ L(t1)/2.

By definition of σ4 either there holds σ2 ≥ σ3 or σ2 ≥ L(t1)2/8.If σ2 ≥ σ3 by definition of σ3 it immediately follows L(t1)/2 ≤ m(σ2). In

the case σ2 ≥ L(t1)2/8 the corollary follows by the estimate

L(t1)/2 = L(t1)1/3(L(t1)2/8)1/3 ≤ L(t1)1/3σ1/32 .

Lemma 3.21. Let x : S1× [0, T∗)→ Rn be a maximal smooth curve shorteningflow. Then the family x(·, t) converges uniformly to a Lipschitz curve x∗, andthe measures µt on S1 converges to a Radon measure µ∗ as t tends to T∗.

30

Note that µ∗ is not the pull back measure of H1bx∗(S1), but by lower semi-continuity of the length functional there holds L(x∗(I)) ≤ µ∗(I).

Proof. By the compactness of Radon measures there exists a monotone sequenceti T∗ and a Radon measure µ∗ on S1 such that

µ∗ = limi→∞

µti ,

and by the monotonicity from lemma 3.3, the full limit exists

µ∗ = limtT∗

µt = inft<T∗

µt.

By lemma 3.3, the family x(·, t) is uniformly Lipschitz continuous. Hence byArzela Ascoli for every ti T∗ there exists a subsequence x(·, tij ) which con-verges in C0 to a Lipschitz curve x∗.

By corollary 3.20, for every u ∈ S1 and 0 ≤ t1 ≤ t2 < T∗ there holds

|x(u, t1)− x(u, t2)| ≤ 2

∫ t2

t1

∫S1

|k|2 dµtdt+ 3µ0(S1)1/3|t1 − t2|1/3.

Hence the full limit x∗(u) = limtT∗ x(u, t) exists for all u ∈ S1.

3.5 Integral Bounds for the Curvature

The Lp−estimates in this capture are proven by Angenent [5] for some generalflow of curves on surfaces. We specialize these estimates to the curve shorten-ing flow in an Euclidean space, but generalize them to arbitrary co-dimension.The generalization to higher co-dimension is straight forward, but we use theopportunity to clarify the dependency on the parameter p.

We denote the Lp−norm of the curvature by

Kp(t) := ‖k(·, t)‖Lp(µt) :=

(∫S1

|k(u, t)|p dµt(u)

)1/p

.

The results of this capture are the following three theorems, which can becompared with Angenent [5], theorem 7.2, theorem 7.3 and theorem 8.1.

Theorem 3.22. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 1 < p <∞ there holds

Kp(t) ≤(Kp(0)−

2pp−1 − C(p)t

)− p−12p

,

where C(p) := 18(4p/(p−1) + 1).

Theorem 3.23. Let x : S1 × [0, t0] → Rn be a smooth curve shortening flow.Then for all 1 < p <∞ and 0 < t ≤ t0 there holds the curvature bound

supS1

|k(·, t)| ≤ C(p)App−1 t−

12p ,

where C(p) := exp(5p/(p− 1)) and

A := max

t− (p−1)2

2p2

0 , tp−1

2p2

0 supt∈[0,t0]

Kp(t)

.

31

The curvature estimates lead to the following short time existence result forinitial curves with Lp bounded curvature.

Theorem 3.24. Let x0 ∈ H2,p(S1,Rn), 1 < p < ∞ be an immersed curve.Then there exists t∗ > 0, depending only on p and ‖k‖Lp , a continuous mapx : S1 × [0, t∗] → Rn, and a non-decreasing, degree one Lipschitz map φ suchthat the restriction to S1×(0, t∗] is a smooth curve shortening flow and satisfyingthe initial condition x(φ(u), 0) = x0(u), u ∈ S1.

To prove theorem 3.22 and 3.23 we need the following lemma.

Lemma 3.25. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 1 < p <∞ there holds

d

dtKp(t) ≤ −

p− 1

12p2

K2p+1p (t)

K2pp/2(t)

+ C(p)Kp(t)3p−1p−1 ,

where C(p) := 9(4p/(p−1) + 1)(p − 1)/p. For p ≥ 2 this holds in the classicalsense and for 1 < p < 2 the map Kp lies in W 1,∞

loc ([0, T∗)) and the inequalityholds in the weak sense.

We first show that lemma 3.25 implies theorem 3.22.

Proof of theorem 3.22. By lemma 3.25, there holds

d

dtKp ≤ C ′(p)K

3p−1p−1p ,

where C(p) := 9(4p/(p−1) + 1)(p− 1)/p. Integrating this inequality leads to theestimate

Kp(0)−2pp−1 −Kp(t)

− 2pp−1 ≤ 2p

p− 1C(p)t.

Proof of lemma 3.25. We first prove the lemma under the assumption p ≥ 2.In this case the function φ(k) = |k|p is C2. By the evolution equation (12) wehave

(|k|p)t =p

2|k|p−2

((|k|2)ss − 2|ks|2 + 4|k|4

)= (|k|p)ss −

4(p− 2)

p

∣∣∣(|k|p/2)s

∣∣∣2 − p|k|p−2|ks|2 + 2p|k|p+2.

Together with the estimate

|k|2|ks|2 = |k|2|k⊥s |2 + |k|2〈T, ks〉2 ≥ 〈k, ks〉2 + |k|2〈T, ks〉2 =1

4

∣∣(|k|2)s∣∣2 + |k|6

this implies

(|k|p)t ≤ (|k|p)ss −4(p− 1)

p

∣∣∣(|k|p/2)s

∣∣∣2 + p|k|p+2.

Integration of this inequality leads to

d

dt

∫S1

|k|p dµt ≤ −4(p− 1)

p

∫S1

|(|k|p/2)s|2 dµt + (p− 1)

∫S1

|k|p+2 dµt. (30)

To estimate the right hand side, we use the following lemma.

32

Lemma 3.26. For a smooth function f ∈ C∞(S1,R) there holds

‖f‖∞ ≤ |f |+ ‖f‖1/2L2(S1) ‖Df‖1/2L2(S1),

where f := 1/(2π)∫f(u)du.

Proof of lemma 3.26. We first prove the lemma for a function with vanishingaverage,

∫S1 f(u)du = 0. For such a function there exists an u ∈ S1 such that

f(u) = 0, and hence we have the estimate

‖f‖∞ ≤1

2

∫|Df(u)|du.

We actually use this inequality for f2 to obtain

‖f‖2∞ = ‖f2‖∞ ≤1

2

∫|D(f2)|du ≤ ‖f‖L2(S1) ‖Df‖L2(S1).

For general functions without vanishing average, we have the estimate

‖f‖∞ ≤ |f |+ ‖f − f‖∞ ≤ |f |+ ‖f − f‖1/2L2(S1) ‖Df‖1/2L2(S1).

The lemma follows since ‖f − f‖2L2(S1) = ‖f‖2L2(S1) − 2π|f |2.

We use lemma 3.26 to estimate the second integral of the inequality (30)

Kp+2p+2 ≤

∥∥∥|k|p/2∥∥∥4/p

∞‖k‖pLp(µt)

≤(∥∥∥(|k|p/2)s

∥∥∥1/2

L2(µt)

∥∥∥|k|p/2∥∥∥1/2

L2(µt)+ L−1

∥∥∥|k|p/2∥∥∥L1(µt)

)4/p

‖k‖pLp(µt),

where we denote the length by L := L(x(·, t)). For λ ∈ (0, 1] there holds(A + B)λ ≤ Aλ + Bλ, and for λ ≥ 1 we have the convex estimate (A + B)λ ≤2λ−1(Aλ + Bλ). We use the suitable of this two inequalities with λ = 4/p ≤ 4to estimate the first factor and Holder inequality for the second factor

Kp+2p+2 ≤ 8

(∥∥∥(|k|p/2)s

∥∥∥2/p

L2(µt)

∥∥∥|k|p/2∥∥∥2/p

L2(µt)+

1

L2/p

∥∥∥|k|p/2∥∥∥4/p

L2(µt)

)‖k‖pLp(µt)

= 8

(∫S1

|(|k|p/2)s|2 dµt)1/p

‖k‖p+1Lp(µt)

+8

L2/p‖k‖p+2

Lp(µt).

Together with the estimate AλB1−λ ≤ ελA + ε−λ/(1−λ)(1− λ)B with λ = 1/pand ε = 1/4 this implies

Kp+2p+2 ≤

2

p

∫S1

|(|k|p/2)s|2 dµt + C1 ‖k‖p(p+1)p−1

Lp(µt)+

8

L2/p‖k‖p+2

Lp(µt), (31)

where C1 := 8 · 4p/(p−1). By Fenchel’s inequality [24] and Holder’s inequalitythere holds

2π ≤∫S1

|k| dµt ≤ L(p−1)/p ‖k‖Lp(µt).

33

This implies the estimate

1

L2/p‖k‖p+2

Lp(µt)≤ ‖k‖

p(p+1)p−1

Lp(µt),

so the last term in (31) can be absorbed to arrive at

Kp+2p+2 ≤

2

p

∫S1

|(|k|p/2)s|2 dµt + C2 ‖k‖p(p+1)p−1

Lp(µt),

where C2 := 8(4p/(p−1) + 1). Together with inequality (30), it follows

d

dt

∫S1

|k|p dµt ≤ −2(p− 1)

p

∫S1

|(|k|p/2)s|2 dµt + (p− 1)C2 ‖k‖p(p+1)p−1

Lp(µt).

With the notation Kp = ‖k(·, t)‖Lp(µt) this inequality reads

d

dtKpp ≤ −

2(p− 1)

p

∥∥∥(|k|p/2)s

∥∥∥2

L2(µt)+ (p− 1)C2K

p(p+1)p−1

p .

This proves the following lemma.

Lemma 3.27. For p ≥ 2 there holds

d

dtKp ≤ −

2(p− 1)

p2Kp−1p

∥∥∥(|k|p/2)s

∥∥∥2

L2(µt)+ C3K

3p−1p−1p , (32)

where C3 := 8(4p/(p−1) + 1)(p− 1)/p.

It can be proven that |k(·, t)|p/2 ∈ W 1,∞(S1, µt) for 1 < p < 2 and that thelemma above holds in the weak sense for 1 < p < 2, but we do not need thisfact.

For further estimates we need the following interpolation inequality.

Lemma 3.28. There exists C <∞ such that for all f ∈ C∞(S1,C) there holds

‖f‖2L2(S1) ≤ C‖Df‖2/3L2(S1) ‖f‖

4/3L1(S1) + |f | ‖f‖L1(S1),

where f := 1/(2π)∫f(u)du.

Proof of lemma 3.28. We first prove the inequality for function with f = 0. Letf(u) =

∑k∈Z ake

iku be the Fourier series of f . By Parseval’s identity, and theestimate |ak| ≤ ‖f‖L1(S1) we have

‖f‖2L2(S1) =∑k∈Z|ak|2 ≤

∑|k|≤N

|ak|2 +1

(N + 1)2

∑|k|>N

|kak|2

≤ 2N‖f‖2L1(S1) +1

(N + 1)2‖Df‖2L2(S1).

Note that if ‖Df‖2L2(S1) = 0, then |ak|2 = 0 for all k 6= 0, and since by assump-tion a0 = 0 the lemma is trivially true. Hence we may minimize the right handside with the choice of N

N <

(‖Df‖2L2(S1)

‖f‖2L1(S1)

)1/3

≤ N + 1,

34

and this leads to the estimate

‖f‖2L2(S1) ≤ 3‖Df‖2/3L2(S1)‖f‖4/3L1(S1).

For a general function f ∈ Cω(S1,C) there holds

‖f‖2L2(S1) ≤ ‖f − f‖2L2(S1) + |f | ‖f‖L1(S1),

and

‖f − f‖L1(S1) ≤ ‖f‖L1(S1) +

∣∣∣∣∫S1

f(x) dx

∣∣∣∣ ≤ 2‖f‖L1(S1).

Together with the result for g = f − f this implies

‖f‖2L2(S1) ≤ 6‖Df‖2/3L2(S1)‖f‖4/3L1(S1) + |f | ‖f‖2L1(S1).

By lemma 3.28 and the estimate (A+B)3 ≤ 4(A3 +B3) we have∥∥∥|k|p/2∥∥∥6

L2(µt)≤ 24

∥∥∥(|k|p/2)s

∥∥∥2

L2(µt)

∥∥∥|k|p/2∥∥∥4

L1(µt)+ 4L−3

∥∥∥|k|p/2∥∥∥6

L1(µt),

where L := L(x(·, t)). With our notation this is

K3pp ≤ 24

∥∥∥(|k|p/2)s

∥∥∥2

L2(µt)K2pp/2 + 4L−3K3p

p/2.

Together with inequality 32 this implies

d

dtKp ≤ −

2(p− 1)K2p+1p

24p2K2pp/2

+2(p− 1)Kp

p/2

6p2L3Kp−1p

+ C3K3p−1p−1p ,

where C3 := 8(4p/(p−1) + 1)(p− 1)/p. By Holder inequality there holds Kpp/2 ≤

LKpp and by Fenchel’s inequality we have 2π ≤ L(p−1)/pKp. This allows us to

estimate the termKpp/2

L3Kp−1p

≤ Kp

L2≤ K

3p−1p−1p ,

and finitely obtain the inequality

d

dtKp ≤ −

p− 1

12p2

K2p+1p

K2pp/2

+ C4K3p−1p−1p , (33)

where C4 := 9(4p/(p−1) + 1)(p− 1)/p. This finishes the prove in the case p ≥ 2.For 1 < p < 2 the function |k|p is only C1 in k. As in lemma 3.4, we use the

function φε(k) = (|k|2 + ε2)1/2 instead of |k|, and compute

(φpε)t = (φpε)ss −p(p− 2)

4φp−4ε

∣∣(|k|2)s∣∣2 − pφp−2

ε |ks|2 + 2pφp−2ε |k|4

≤ (φpε)ss −p(p− 1)

4φp−4ε

∣∣(|k|2)s∣∣2 + pφp−2

ε |k|4

≤ (φpε)ss −4(p− 1)

p

∣∣∣(φp/2ε )s

∣∣∣2 + pφpε |k|2

35

By integration this leads to the inequality

d

dt

∫S1

φpε dµt ≤ −4(p− 1)

p

∫S1

|(φp/2ε )s|2 dµt + (p− 1)

∫S1

φp+2ε dµt,

which corresponds to the inequality (30). By the same argument as above, withφε(k) in place of |k|, we obtain

d

dtXp,ε ≤ −

p− 1

12p2

X2p+1p,ε

X2pp/2,ε

+ C4X3p−1p−1p,ε , (34)

where Xp,ε(t) := ‖φε(k(·, t))‖Lp(µt) and C4 is the same constant as in inequality(33). The map φε(k) converges to |k| in C0(S1) as ε tends to zero. Therefore,Xp,ε(t) converges to Kp(t) as ε tends to zero.

Now we claim Kp ∈ W 1,∞loc ([0, T∗)). This follows if we can find a Lipschitz

bound for Xp,ε which is independent of ε. At a fixed time t we compute∣∣∣∣ ddtXpp,ε

∣∣∣∣ =

∣∣∣∣∫S1

p〈k, kt〉φpp

− |k|p+2 dµt

∣∣∣∣ ≤ 2π supS1

p|k|p−1|kt|+ |k|p+2.

Together with Fenchel’s and Holder’s inequality this implies∣∣∣∣ ddtXp,ε

∣∣∣∣ =1

pXp−1p,ε

∣∣∣∣ ddtXpp,ε

∣∣∣∣ ≤ (2π)2−pL(p−1)2/p supS1

|k|p−1|kt|+|k|p+2

p.

where L is the length of x(·, t). Hence for each fixed t0 ∈ [0, T∗), the map Kp isLipschitz in [0, t0] with Lipschitz constant

C(p, t0) = (2π)2−pL(0)(p−1)2/p supS1×[0,t0]

|k|p−1|kt|+|k|p+2

p,

where L(0) is the length of x(·, 0). By Rademacher Kp(t) is L1 almost every-where differentiable in [0, T∗) and lemma 3.25 follows by inequality (34).

Proof of theorem 3.23. By scaling xλ(u, t) = λx(u, t/λ2) with λ :=√t0, it is

enough to prove the theorem in the case t0 = 1. We assume that Kp/2(t) ≤ At−αholds for all t ∈ (0, 1], for some fixed A, α and p. The goal is to find suitableB and β such that Kp(t) ≤ Bt−β holds for all t ∈ (0, 1]. This follows by themaximum principle, if we can show that Yp(t) := Bt−β satisfies the reverseinequality of lemma 3.25, i.e.

d

dtYp ≥ −λ

Y 2p+1p

Y 2pp/2

+ C4Y3p−1p−1p ,

where λ := (p− 1)/(12p2) and C4 := 9(4p/(p−1) + 1)(p− 1)/p. This leads to theinequality

−βBt−β−1 ≥ −λB2p+1

A2pt2pα−(2p+1)β + C4B

3p−1p−1 t−

3p−1p−1 β .

This condition can be rewritten as

β ≤ λB2p

A2pt1−2p(β−α) − C4B

2pp−1 t1−

2pp−1β .

36

With the choice β = α+ 1/(2p), the exponent 1− 2p(β−α) = 0 vanishes. If weassume β ≤ (p− 1)/2p, the last exponent of t is positive and it suffices to prove

βB2pp−1 + C4B

2pp−1 ≤ λB

2p

A2p.

If we additionally assume B ≥ 1 an even stricter condition is

C5B2pp−1 ≤ λB

2p

A2p,

where C5 := 10(4p/(p−1) + 1)(p − 1)/p. Therefore, the inequality Kp ≤ Bt−β ,t ∈ (0, 1] holds for

B =

(C5

λ

) p−12p(p−2)

Ap−1p−2

=(

120(4p/(p−1) + 1)p) p−1

2p(p−2)

Ap−1p−2 . (35)

Now, we iterate this procedure to obtain sequences pj , Aj and αj such thatKpj (t) ≤ Ajt−αj holds for all t ∈ (0, 1]. For a fixed p0 > 1, we define pj := 2jp0,α0 = 0, and αj+1 := αj + 1/(2pj) = (1− 2−j)/(2p0). We verify the assumptionsαj ≤ 1/(2p0) ≤ 1/2 and

αj+1 ≤p0(1− 2−j)

2p0=pj − 1

2pj.

We defineA0 := max1, sup

t∈[0,1]

Kp0(t)

and Aj inductive by the equation (35). Note that the sequence Aj is monotoni-cally increasing and hence there holds Aj ≥ A0 ≥ 1. By definition, the sequenceAj satisfies the logarithmic equation

log(Aj+1) =pj+1 − 1

pj+1 − 2

[log(Aj) +

1

2pj+1log(

120(4pj+1/(pj+1−1) + 1)pj+1

)]Since pj+1 = 2j(2p0) > 2 for all j ≥ 0, we get the estimate

log(Aj+1) ≤ κj+1

[log(Aj) +

2−j

4p0log(240(42 + 1)2jp0

)]≤ κj+1

[log(Aj) + (3 + j)2−j

],

where κj := (pj − 1)/(pj − 2). By the convergence

n∏j≥1

κj =

n∏j≥1

2jp0 − 1

2(2j−1p0 − 1)

=2np0 − 1

2n(p0 − 1)→ p0

p0 − 1, (n→∞)

37

we can further estimate

log(An) ≤

n∏j≥1

κj

log(A0) + 3

n−1∑j=1

2−j

+

n−1∑j=1

j2−j

≤ p0

p0 − 1

[log(Aj) + 3

(1− 2−(n−1)

)+(

2− (n+ 1)2−(n−1))]

≤ p0

p0 − 1[log(Aj) + 5] ,

and concludelimj→∞

Aj ≤ C(p0)Ap0/(p0−1)0 ,

where C(p0) := exp(5p0/(p0 − 1)). This proves the curvature bound

‖k(·, t)‖∞ = limp→∞

Kp(t) ≤ C(p0)Ap0/(p0−1)0 t−1/(2p0),

for all t ∈ (0, 1].

Proof of theorem 3.24. We approximate the initial curve x0 in H2,p(S1,Rn) bysmooth curves xi0 ∈ C∞(S1,Rn). Since x0 is an immersion and S1 is compact,there exists a lower bound infS1 |Dux0| > 0. Hence we may assume that eachxi0, i ≥ 1 is an immersion. By the short time existence theorem 2.3, thereexist maximal smooth curve shortening flows xi : S1 × [0, Ti)→ Rn with initialconditions xi(·, 0) = xi0.

The first goal is to find a lower bound 0 < t∗ ≤ limTi. Since the sequencexi(·, 0) → x0 converges in H2,p, the Lp norm of their curvatures is uniformly

bounded, i.e. K(i)p (0) := ‖ki(·, 0)‖Lp ≤ C for all i ≥ 1. By corollary 3.22 there

exists t∗ > 0 such that for all i ≥ 1 and t ∈ [0, 2t∗) ∩ [0, Ti) there holds

K(i)p (t) ≤ C(2t∗ − t)−

p−12p .

Hence, by theorem 3.23, for all i ≥ 1 there holds

|ki(u, t)| ≤ Ct−12p , (36)

for all u ∈ S1 and t ∈ [0, t∗] ∩ [0, Ti). Clearly, this implies the lower boundTi ≥ t∗.

We fix δ > 0. By theorem 3.6 all arc-length derivatives of the curvature areuniformly bounded on S1 × [δ, t∗] and i ≥ 1. More precisely, for every l ≥ 0there exists a constant Cl depending only on l such that

|Dlski(u, t)| ≤ Clδ

− 12p−

l2 , (37)

holds for all u ∈ S1, t ∈ [δ, t∗] and i ≥ 1. Now we parametrize by constantspeed at time t = t∗/2. There exist diffeomorphisms φi : S1 → S1 such thatxi(v, t) := xi(φ

−1i (v), t) satisfies

|(xi)v(u, t∗/2)| = ci :=L(xi(·, t∗/2))

2π.

38

The maximum principle with a ball with radius L(xi(·, t∗/2)) implies that√t∗/2 ≤ L(xi(·, t∗/2)). Therefore, there holds√

t∗/2

2π≤ ci ≤

L(xi(·, 0))

2π. (38)

In particular, this implies an uniform bound for |(φi)u| ≤ Lip(x0)/ci. By (37)and lemma 3.7, for each (m, l) ∈ N2\0 there exists a constant Km;l dependingonly on (m, l), δ, t∗ and p such that

|Dmt D

lvx(v, t)| ≤ c−li Km;l, (v, t) ∈ S1 × [δ, t∗].

By the lower bound (38), the right hand side is bounded independently of i.Since this holds for all δ > 0, there exist a subsequence ij , a smooth map x :

S1 × (0, t∗]→ Rn a non-decreasing, and degree one Lipschitz map φ : S1 → S1

such thatxij → x, in C∞loc(S

1 × (0, t∗]),

φij → φ, in C0(S1).

Note that x is a smooth curve shortening flow on S1 × (0, t∗].It is left to prove that the limit x(·, 0) := limt→0 x(·, t) exists and satisfies

x(φ(u), 0) = x0(u). By the curvature bound (36) we have∫ t

0

|xt(u, τ)|dτ ≤ C∫ t

0

dτ

τ1/(2p)≤ Ct

2p−12p .

Therefore, the limit x(·, 0) exists and we can estimate

|x(φ(u), 0)− x0(u)| ≤ |x(φ(u), 0)− x(φ(u), t)|+ |x(φ(u), t)− xij (φ(u), t)|+|xij (u, t)− xij0 (u)|+ |xij0 (u)− x0(u)|.

Again by the curvature bound (36), the third term is bounded by Ct(2p−1)/2p.Hence in the limit j →∞ there holds

|x(φ(u), 0)− x0(u)| ≤ Ct2p−12p .

In the limit t→ 0 this proves the desired initial value condition.

3.6 Graph Flow

Definition 3.29. Let [a, b] ⊂ R be a compact interval. A smooth functionr : [a, b]× [0, T∗)→ Rn−1, (z, t) 7→ r(z, t), is called a graph flow, if it solves theequation

rt =rzz

1 + |rz|2, (z, t) ∈ (a, b)× [0, T∗). (39)

By proposition 2.2, the motion of the graph x(z, t) = (z, r(z, t)) differs fromthe curve shortening flow only by a tangential motion. The advantage of equa-tion (39) is, that it is strictly parabolic as long as the first derivative rz isbounded. In fact, if |rz| ≤ 1/10, we will obtain local estimates of the secondderivative. The following theorem is from Altschuler and Grayson [8]. Note thatthey state it only for n = 3, but the proof for general n ≥ 3 is identical. Wereproduce their proof for convenience of the reader. In [20], Ecker and Huiskenestablishes similar local estimates for hypersurfaces that are graphs.

39

Theorem 3.30. (Altschuler-Grayson [8], Theorem 4.3) Let r(z, t) be a solutionto the graph flow for (z, t) ∈ Ω := [0, δ]× [0, T∗). Assume that |rz| ≤ 1/10 holdson Ω. Then the curvature k(·, t) of the graph of r(·, t) satisfies

|k(z, t)|2 ≤ 1

t+

δ2

z2(δ − z)2,

for (z, t) ∈ Ω.

Proof. Since we assume that |rz| < 1 we may consider the quantity

Q :=|rt|2

1− |rz|2,

and its evolution equation

Qt =Qzz

1 + |rz|2− 8 〈rtz, rt〉〈rt, rz〉

(1− |rz|2)2− 8|rt|2 〈rt, rz〉2 (1 + |rz|2)

(1− |rz|2)3

− 2|rtz|2

(1 + |rz|2)(1− |rz|2)− 4|rt|2 〈rtz, rz〉

(1 + |rz|2)(1− |rz|2)

− 2|rt|4(1 + |rz|2)

(1− |rz|2)2− 4|rt|2 〈rt, rz〉2

(1− |rz|2)2.

Provided that |rz| < 1/10 there holds

Qt ≤Qzz

1 + |rz|2−Q2.

Define f(z) := z(δ − z) and g(z) := δ2/f(z)2. A simple calculation shows thatgzz(z) ≤ g(z)2 for z ∈ (0, δ). With h(z, t) := 1/t+ g(z) we have

(Q− h)t ≤Qzz

1 + |rz|2−Q2 +

1

t2+g2 − gzz1 + |rz|2

≤ Qzz − gzz1 + |rz|2

−Q2 + (1/t+ g)2

≤ (Q− h)zz1 + |rz|2

− (Q− h)(Q+ h).

Note that (Q−h)(z, t)→ −∞ as (z, t) tends to the parabolic boundary of Ω.For each t ∈ [0, T∗) we have the upper bound (Q−h)(·, t) ≤ sup[0,δ]Q(·, t) <∞.Hence for all t ∈ (0, T∗), the maximum max[0,δ](Q − h)(·, t) is attained in theinterior of [0, δ], and the maximum principle implies Q−h ≤ 0 on [0, δ]× [0, T∗).

Since the curvature k of the graph is the normal component of (0, rt), thisimplies the estimate

|k|2 ≤ |rt|2 = Q(1− |rz|2) ≤ Q ≤ h,

on [0, δ]× [0, T∗).

Since the graph flow equation (39) is strictly parabolic we can deduce interiorbounds for higher derivatives by standard theory.

40

Lemma 3.31. Let r : Ω := [a, b]× [0, T∗)→ Rn−1 be a graph flow with |rz| ≤ η.Assume the curvature k of the graph x(z, t) := (z, r(z, t)) satisfies |k| ≤ M onΩ. For all m ≥ 1 and δ > 0 there exists a constant Cm depending only on M ,η, δ and m such that ∑

1≤i+2j≤m

∣∣∣DizD

jt r∣∣∣0;Q≤ Cm, (40)

where Q := [a+ δ, b− δ]× [δ, T∗).

Proof. Since r is a graph flow, the curvature k is the normal projection of (0, rt),i.e.

k = (0, rt)⊥ := (0, rt)− 〈(0, rt), T 〉T,

where T := (1, rz)/√

1 + |rz|2 is the tangent vector. This implies the point-wiseestimate

|rt|√1 + |rz|2

≤ |k| ≤ |rt|.

By the slope bound in particular there holds |rt|0;Ω ≤M√

1 + η2.Our next goal is to apply the interior Schauder estimate to the evolution

equation∂

∂trz =

∂

∂z

(1

1 + |rz|2∂

∂zrz

). (41)

For this reason we need a Holder bound for the coefficient (1 + |rz|2)−1. Such aHolder bound follows either by the evolution equation

∂

∂t(1 + |rz|2) =

∂

∂z

(1

1 + |rz|2∂

∂z(1 + |rz|2)

)− 2|rt|2(1 + |rz|2), (42)

or

∂

∂t

√1 + |rz|2 =

∂

∂z

(1

1 + |rz|2∂

∂z

√1 + |rz|2

)− |k|2

√1 + |rz|2. (43)

Both equations are strictly parabolic and has bounded coefficients. We use (42).By Lieberman [43] theorem 6.29, for each α ∈ (0, 1) the Holder semi-norm isbounded by

[1 + |rz|2]α;Q ≤ C(M,η, α, δ)|1 + |rz|2|0;Ω. (44)

Since |rz|0;Ω ≤ η, the right hand side of (44) is bounded by a constant K, thatonly depends on M , η, α and δ. Since the derivative of the function f(x) = x−1

is bounded by one for x ≥ 1, with the same constant K there holds

[(1 + |rz|2)−1]α;Q ≤ K(M,η, α, δ).

Now we can apply the interior Schauder estimate, see for example Lieberman[43] theorem 4.9, to the evolution equation (41) to conclude

|rz|2+α;Q ≤ C(M,η, δ,K)|rz|0;Ω,

and by a bootstrap argument we obtain

|rz|m+α;Q ≤ C(M,η,m+ α, δ)|rz|0;Ω, (45)

41

where|f |m+α;Q :=

∑i+2j≤m

|DizD

jt f |0;Q +

∑i+2j=m

[DizD

jt f ]α;Q.

Since |rz|0;Ω ≤ η, this shows that for each m ≥ 1 there exists a constant Cdepending only on M , η m and δ such that there holds

|DizD

jt r|0;Q ≤ C(M,η,m, δ), (46)

for all (i, j) with i+ 2j ≤ m and i ≥ 1.

Recall that we already have shown |rt|0;Ω ≤ M√

1 + η2. It is left to prove

the lemma for the derivatives |Djt rt| for j ≥ 1. Since r is a graph flow there

holds

Djt rt = Dj

t

(rzz

1 + |rz|2

)=

j∑i=0

(j

i

)(Dit

1

1 + |rz|2

)(Dj−it rzz

).

The right hand side is already bounded by inequality (46), and it follows

|Djt rt|0;Q ≤ C(M,η, 2j + 2, δ). (47)

Inequality (40) follows by (46) and (47).

As mentioned above, the graph x(z, t) = (z, r(z, t)) of a graph flow r(z, t) isup to a time depending diffeomorphism equal to a curve shortening flow x(u, t).Note that the domain of x(·, t) depends on t. Under a curvature bound it ispossible to get a curve shorting flow on a time independent domain for a shorttime interval.

Lemma 3.32. Let r : I × [0, t∗] → Rn−1 be a graph flow, where I = [a, b] ⊂ Rbe a compact interval. Assume |rz| ≤ η and |k| ≤ C0 hold on I × [0, t∗], wherek denotes the curvature of the graph of r(·, t). Fix σ ∈ (0, (b− a)/2) and defineδ := σ/C0. Then for each t0 ∈ [0, t∗ − δ] there exists a smooth family ofdiffeomorphisms ψ(·, t) : V → ψ(V ) ⊂ I, t ∈ [t0, t0+δ], where V := [a+σ, b−σ],such that ψ(u, t0) = u and

x(u, t) := (ψ(u, t), r(ψ(u, t), t)), (u, t) ∈ V × [t0, t0 + δ]

defines a smooth curve shorting flow, i.e. x(·, t) is an immersion for everyt ∈ [t0, t0 + δ] and there holds xt = k on V × [t0, t0 + δ].

Moreover, there holds√

1 + η2 ψu ≥ |xu| ≥ e−C0σ on V × [t0, t0 + δ].

Proof. We may assume that t0 = 0. For each u ∈ V , we define the mapψ(u, ·) : [0, δ]→ I, by ψ(u, 0) = u and

∂

∂tψ(u, t) = −〈rz(ψ(u, t), t), rt(ψ(u, t), t)〉

1 + |rz(ψ(u, t), t)|2.

Note that the right hand side is defined as long as ψ(u, t) ∈ I. Since

〈rz, rt〉1 + |rz|2

≤ |rz| |k|√1 + |rz|2

≤ |k| ≤ C0,

42

there holds |ψ(u, t) − ψ(u, 0)| ≤ C0t ≤ σ for all 0 ≤ t ≤ σ/C0 = δ. Sinceψ(u, 0) = u ∈ V , this shows that ψ is well defined.

The map x : V × [0, δ]→ Rn defined by

x(u, t) := (ψ(u, t), r(ψ(u, t), t)), (u, t) ∈ [a, b]× [0, t∗].

satisfiesxt = ψt(1, rz) + (0, rt).

By multiplication of the tangent vector T = (1, rz)/√

1 + |rz|2, we observe

〈xt, T 〉 = ψt√

1 + |rz|2 + 〈rz, rt〉 = 0.

Therefore, x satisfies the standard curve shortening flow equation xt = k. Bylemma 3.3 there holds

|xu(u, t)| ≥ e−C20 t |xu(u, 0)|.

Since xu = ψu(1, rz) we have |xu(u, 0)| =√

1 + |rz(u, 0)|2 ≥ 1. This leads tothe lower bound√

1 + η2|ψu(u, t)| ≥ |xu(u, t)| ≥ e−C0σ, (u, t) ∈ V × [0, δ].

Since ψu(u, 0) = 1 this implies ψu(u, t) ≥ e−C0σ/√

1 + η2. For each t ∈ [0, δ],the map ψ(·, t) : V → ψ(V, t) ⊂ I is a diffeomorphism and x(·, t) is a immersion.

3.7 Local Lipschitz Graphs

An oriented curve is an equivalence class of continuous map x : S1 → Rn, mod-ulo orientation preserving homeomorphism of S1. With this definition a curveis rectifiable if and only if there exists a Lipschitz continuous representative. Bya Lipschitz curve we always understand such a Lipschitz continuous map. Foran absolutely continuous curve x : S1 → Rn, the derivative xu(u) exists almosteverywhere, and we can define the arc-length parametrization in the sense that|xu(u)| = 1 for almost every u. Such a parametrization is clearly Lipschitzcontinuous. Hence in the equivalence class there is no difference between abso-lutely continuous and Lipschitz curves. Clearly, graphs of absolutely continuousfunctions are much more general than graphs of Lipschitz functions.

Definition 3.33. Let U be either [0, 1] or S1. A Lipschitz curve x : U → Rn iscalled a Lipschitz ε−graph with Lipschitz constant Lipε(x), if every segment oflength less than ε is the graph of a Lipschitz function r : V ⊂ L→ L⊥ over a lineL, with Lipschitz constant Lip(r) ≤ Lipε(x). i.e. for every open interval I ⊂ S1

such that L(x(I)) ≤ ε, there exist a unit vector v, an open interval V ⊂ Rand a monotone non-decreasing Lipschitz map φ : V → I and a Lipschitz mapr : V → L⊥, where L = span(v), such that there holds x(φ(z)) = zv + r(z) forall z ∈ V .

A curve is locally Lipschitz graph-like, if the curve is a Lipschitz ε−graph forsome ε > 0.

43

Up to rotation, we may always assume that the unit vector is v = (1, 0, ..., 0)such that we can write x(φ(z)) = (z, r(z)). If the Lipschitz ε−graph is para-metrized by arc-length, then the Lipschitz map φ : V → I is strictly monotone.

The definition above is related to Angenent’s work [5, 6], where he studies amore general flow of closed immersed curves on surfaces. The following definitionis used by Angenent, and is just translated in to our setting.

Definition 3.34. Let U be either [0, 1] or S1. For an absolutely continuouscurve x : U → Rn, we define the angle

αε(x) := infα > 0 : 〈xu(u1), xu(u2)〉 ≥ |xu(u1)| |xu(u2)| cos(α),

for a.e. u1, u2 ∈ U with L(x|[u1, u2]) ≤ εand xu(u1) 6= 0 6= xu(u2).

The angle αε(x) measures the maximal angle between two tangent vectors,at points with intrinsic distance less than ε. Recall that the intrinsic distancebetween u1 and u2 is the length of the segment x|[u1, u2],

L(x|[u1, u2]) :=

∫ u2

u1

|xu(v)|dv.

Note that the map x : U → Rn is C1 if and only if limε→0 αε(x) = 0.Since our work is also influenced by the work of Altschuler and Grayson [8]

on ramps, we mention their nomenclature. They call a curve ε−flat, ε−very flat,or ε−extremely flat if the angle αε less than 1/10, 1/100, or 1/1000, respectively.

A difference between curves on surfaces and curves in arbitrary co-dimensionis that the curvature does have a sign with respect to a chosen unit normal vectorν. For C2 curves in R2 we have the equality

αε(x) = sup

∣∣∣∣∫ u2

u1

kscdµ

∣∣∣∣ : L(x|[u1, u2]) ≤ ε,

where k = kscν, whereas for C2 curves in Rn there only holds the inequality

αε(x) ≤ sup

∫ u2

u1

|k|dµ : L(x|[u1, u2]) ≤ ε.

Related to this fact, the relation between Lipε(x) and αε(x) is slightly moreinvolved in higher co-dimension. In co-dimension one, under the conditionαε(x) < π there holds

Lipε(x) = tan

(αε(x)

2

).

The following proposition shows the relation between αε(x) and Lipε(x) inarbitrary co-dimension.

Proposition 3.35. Let x : S1 → Rn be a Lipschitz curve. If

sin

(αε(x)

2

)<

√n

2(n− 1), (48)

then the curve is a Lipschitz ε−graph, and there holds

Lipε(x) ≤

√(n− 1)(1− cos(αε(x)))

1 + (n− 1) cos(αε(x)). (49)

44

Conversely, for a Lipschitz ε−graph there holds

tan

(αε(x)

2

)≤ Lipε(x). (50)

Remark. We computed the optimal dimension depending constants only to clar-ify the difference to the co-dimension one case. Later we will use proposition3.35 only with the dimension independent condition

sin

(αε(x)

2

)<

1√2,

which is equivalent to αε(x) < π/2 instead of (48) and the dimension indepen-dent estimate

Lipε(x) ≤ tan(αε(x)),

instead of (49).

Proof. We first prove the second part of the proposition. Let x(z) = (z, r(z))be the graph of a Lipschitz map r : I ⊂ R → Rn−1. By the Lipschitz boundthere holds |rz| ≤ Lip(r) and |xz| ≤

√1 + Lip(r)2 for a.e. z ∈ I. For the angle

α between xz(z1) and xz(z2) we have

(1 + Lip(r)2) cos(α) ≥ 〈xz(z1), xz(z2)〉 = 1 + 〈rz(z1), rz(z2)〉 ≥ 1− Lip(r)2,

and this implies tan(α/2) ≤ Lip(r). Since this holds all z1, z2 such that xz(z1)and xz(z2) exists, inequality 50 follows.

The first part of the proposition follows directly from the next lemma.

Lemma 3.36. Let x : I → Rn be a Lipschitz curve, define on a compact intervalI ⊂ R and assume |xu(u)| = 1 for a.e. u ∈ I. If the supremum angle betweentwo tangent vectors α := ess sup∠(xu(u1), xu(u2)) : u1, u2 ∈ I ∈ [0, π] satisfies

sin(α/2) <

√n

2(n− 1), (51)

then there exists a line L in Rn and an interval [a, b] ⊂ L and a functionr : [a, b]→ Rn such that the image of x is the graph of r and there holds

Lip(r) ≤

√(n− 1)(1− cos(α))

1 + (n− 1) cos(α)≤ tan(α).

Proof. Since x is Lipschitz the derivative exists almost everywhere. There existsN ⊂ I with µ(N) = 0 such that the tangent vector T (u) := xs(u) exists for allu ∈ I \N and we denote the image the tangent vectors in Sn−1 by

K := T (I \N)).

By definition of α is the essential diameter of the image of the tangent vectors,i.e.

α = α(K) := supdSn−1(p, q) : p, q ∈ K.

A curve x is the graph of a Lipschitz function r if and only if the image of thetangent vectors K is contained in a ball Bβ(q) of Sn with radius β < π/2. In

45

this case, the curve can be written as graph over the line span(q) and thereholds the Lipschitz bound Lip(r) ≤ tan(β). We define the circumcircle radius

β = β(K) := infr > 0 : K ⊂ Br(q), q ∈ Sn−1,

where Br(q) := p ∈ Sn : dSn−1(p, q) ≤ r.Hence we only have to relate the diameter α to the circumcircle radius β of

a compact set K in Sn−1 and claim that there holds

sin(β) ≤√

2(n− 1)

nsin(α/2). (52)

We first show that the inequality (52) implies the lemma. The curve x canbe written as graph if and only if β < π/2 which holds if

sin(α/2) <

√n

2(n− 1),

and under this assumption we calculate

Lip(r) ≤ tan(β) ≤

√2(n− 1) sin(α/2)

n− 2(n− 1) sin(α/2)=

√(n− 1)(1− cos(α))

1 + (n− 1) cos(α).

Before we prove inequality (52), we mention here the co-dimension one caseand the dimension independent estimate. In the case n = 2 there holds 2β = α.In this case, by the argument from the last paragraph, for α < π there holdsLip(r) ≤ tan(α/2). For arbitrary dimension n ≥ 2, we use the fact that thecircumcircle radius is bounded by the diameter, i.e. β ≤ α. Hence for α < π/2there always holds Lip(r) ≤ tan(α).

Claim. To prove inequality (52) it is enough to consider n equidistant pointsK = p1, ..., pn in Sn−1.Proof. By compactness of Sn−1 the infimum in the definition of the circum-circle radius is attained with some closed ball Bβ(q). By the definition of βand induction on the dimension n, we may assume that there exist n pointsp1, p2, ..., pn ∈ K ∩ ∂Bβ(q). In fact, there exists a space E of dimension n′ ≤ nthrough q such that there exists n′ points in E ∩K ∩ ∂Bβ(q). By induction onthe dimension we can assume that n′ = n. The distances between each pair ofthe n points is bounded by the diameter α, and under this restriction β attendsits maximum if all distances are equal to α.

Now we will reduce the problem from Sn−1 to Rn−1. We define the Euclideandiameter

d = d(K) := supdRn−1(p, q) : p, q ∈ K,

and circumcircle radius

ρ = ρ(K) := infr > 0 : K ⊂ Br(q), q ∈ Rn−1,

where Br(q) denotes the Euclidean ball in Rn−1.Claim. Inequality (52) holds for all n equidistant points in Sn−1, if and only

if the inequality

ρ ≤√n− 1

2nd, (53)

46

holds for all n equidistant points in Rn−1.Proof. By the last claim we can consider n points K = p1, ..., pn in Sn−1

which have constant distance α to each other. Hence K is contained in theintersection of Sn−1 with a (n− 1)−dimensional affine space E. Moreover, theEuclidean distance dE(pi, pj) between these points is also constant. The relationbetween the Euclidean diameter d and the spherical diameter α is given by

d = 2 sin(α/2), (54)

and the relation between the Euclidean circumcircle radius ρ and the sphericalcircumcircle radius β is

ρ = sin(β). (55)

The claim follows by these two relations.It is left to prove inequality (53) in the case of n equidistant points K =

p1, ..., pn in Rn−1. Proof. By translation and scaling we may assume that∑ni=1 pi = 0 and d = 1. In this case there holds |pi| = ρ for all i = 1, ..., n and

〈pi − pk, pj − pk〉 = 1/2 for all i 6= j 6= k 6= i. Hence we have

n2ρ2 = |npn|2 =

∣∣∣∣∣n−1∑i=1

(pi − pn)

∣∣∣∣∣2

=

n−1∑i=1

|pi − pn|2 +∑i6=j

〈pi − pn, pj − pn〉

= (n− 1) +(n− 1)(n− 2)

2=

(n− 1)n

2.

The next goal is to show that an Lp bound on the curvature, for 1 < p <∞implies that the curve is Lipschitz ε−graph for sufficiently small ε. This willallows us to apply the estimates from section 3.5 to locally Lipschitz graph-likecurves. For this purpose we need the following

Lemma 3.37. Let x ∈ C2(I,Rn) be a curve, defined on a compact intervalI ⊂ R, and assume that its total absolute curvature satisfies

K :=

∫I

|k|dµ < π.

Then the curve is the graph of a function r : [a, b] ⊂ L→ L⊥ and there holds

Lip(r) ≤ tan(K/2).

Proof. We assume that x is parametrized by arc-length and that I = [0, L].There exists u0 ∈ I such that∫ u0

0

|k|dµ =

∫ L

u0

|k|dµ.

By the arc-length parametrization, the tangent vector T := xu ∈ Sn−1 satisfiesTu = k. Hence the total absolute curvature measures the length of the image ofthe tangent vectors in Sn−1,

H1(T ([0, u0])) =

∫ u0

0

|k|dµ =K

2,

47

and analogously for the interval [u0, L]. Therefore, the angle β between T (u0)and T (u) is bounded by β ≤ K/2 < π/2. Hence the curve x is the graph of afunction r over the line L := span(T (u0)) and there holds

Lip(r) ≤ tan(K/2).

Corollary 3.38. Let x ∈ C2(I,Rn) be a curve, defined on a compact intervalI ⊂ R and let Kp := ‖k‖Lp , 1 ≤ p < ∞. Then for all 1 < p < ∞ andε = ε(p,Kp) with Kpε

(p−1)/p < π there holds

Lipε(x) ≤ tan

(Kpε

p−1p

2

).

Proof. We fix J ⊂ S1 with L(x(J)) ≤ ε. By Holder’s inequality there holds∫J

|k|dµ ≤(∫

J

|k|pdµ)1/p

ε(p−1)/p ≤ Kpε(p−1)/p.

By lemma 3.37 the curve segment x(J) is the graph of a function r. Since thisholds for every segment of length less than ε, the curve is a Lipschitz ε−graphand there holds

Lipε(x) = tan (K1/2) ≤ tan

(Kpε

p−1p

2

).

In the next lemma we show that the property that a curve segment can bewritten as Lipschitz graph is preserved under the flow as long as the boundarypoints do not move to far. Note that this estimate is pseudo-local in the sensethat we have to assume the flow is global, i.e. is defined on S1.

Lemma 3.39. Let x : S1× [0, T∗)→ Rn be a smooth curve shortening flow andI = (u1, u2) ⊂ S1, [t0, t1] ⊂ [0, T∗) be intervals. Assume x(I, t0) is the graph ofa function r0 : [a, b] ⊂ L → L⊥ with Lip(r0) ≤ η and the boundary points arecontained in some balls x(uj , t) ∈ Bσ(pj), j = 1, 2 for all time t ∈ [t0, t1].

Then for each t ∈ [t0, t1] the segment x(J(t), t) is the graph of a Lipschitzfunction r(·, t) : V ⊂ L→ L⊥ with Lip(r(·, t)) ≤ 10η, where V := [a+∆, b−∆],J(t) := u ∈ I : πL(x(u, t)) ∈ V and ∆ := σ/(2 minη, 1/10). More precisely,the map

ϕ(·, t) : J(t)→ V, ϕ(u, t) := πL(x(u, t))

is bijective and satisfies

(ϕ(u, t), r(ϕ(u, t), t)) = x(u, t),

where we identify L× L⊥ with R× Rn−1.

48

Proof. We fix z ∈ [a+2σ, b−2σ]. We will use theorem 3.11 with a hyperplane Pzwhich includes the point (z, 0) and is perpendicular to L. Since the hyperplanePz separates the ball Bσ(p1) from Bσ(p2) and the map x(·, t)bI is continuousand connects these balls, the number of intersections

iI(Pz, t) := #u ∈ I : x(u, t) ∈ Pz,

is at least one. Since x(I, t0) is a graph, there holds iI(Pz, t0) = 1. Sincethe boundary points of x(I, t) for t ∈ [t0, t1] are disjoint from the hyperplane,theorem 3.11 implies iI(Pz, t) = 1 for all t ∈ [t0, t1]. Hence the restrictionof x(·, t) to u ∈ I : πL(x(u, t)) ∈ [a + 2σ, b − 2σ] is the graph of a mapf(·, t) : [a+ 2σ, b− 2σ]→ L⊥.

To establish a height bound for f(·, t), we use hyperplanes P which do notseparate the ball Bσ(p1) from Bσ(p2), and do not intersect with them and haveslope tan(∠(L,P )) ≥ η as barriers. Since x(I, t0) is the graph of r0 there holdsP ∩ x(I, t0) = ∅, and by the maximum principle we conclude P ∩ x(I, t0) = ∅for all t ∈ [t0, t1]. This implies the height bounds

|f(z, t)− h1| ≤ η|z − a|+ 2σ, (56)

|f(z, t)− h2| ≤ η|z − b|+ 2σ. (57)

Claim. The restriction r(·, t) = f(·, t)bV satisfies the Lipschitz bound

Lip(r(·, t)) ≤ 10η.

Proof. We define

A := P ⊂ Rn :P is a hyperplane which separates Bσ(p1) from Bσ(p2),

and satisfies tan(∠(L,P )) > 10η .

Again since the map x(·, t) is continuous, for all P ∈ A the number of intersec-tions iI(P, t) is at least one. Since x(I, t0) is the graph of the Lipschitz functionr0, there holds iI(P, t0) = 1 for all P ∈ A. Since the boundary points aredisjoint from the hyperplane P ∈ A, theorem 3.11 implies that the number ofintersections iI(P, t) is non-increasing. Therefore, there holds iI(P, t) = 1 forall t ∈ [t0, t1] and P ∈ A.

Note that the claim follows if we can show that B(u, t) ⊂ A for all t ∈ [t0, t1]and u ∈ J(t), where

B(u, t) := P ⊂ Rn :P is a hyperplane with x(u, t) ∈ P,and tan(∠(L,P )) > 10η .

We fix t ∈ [t0, t1], u ∈ J(t) and P ∈ B(u, t). Let (w, h1) ∈ L × L⊥ be theintersection point between the hyperplane P and the line parallel to L containingp1. By the slope bound tan(∠(L,P )) > 10η the hyperplane P is disjoint fromthe ball Bσ(p1) if

|w − qj | ≥√

1 + (10η)2

10ησ, (58)

where pj = (qj , hj) ∈ L×L⊥, j = 1, 2 are the center-points of the balls Bσ(pj).By the slope bound of P and the height bound (56) there holds

|z − w| ≤ |f(z, t)− h1|10η

≤ |z − a|10

+σ

5η.

49

This leads to the estimate

w − q1 ≥ (z − a)− |z − w| − |q1 − a| ≥ (z − a)− |z − a|10

− σ

5η

≥ 9

10(z − a)− σ

5η.

Hence to prove (58) it is enough to show

9

10(z − a)− σ

5η≥√

1 + (10η)2

10ησ.

By definition of u ∈ J(t) there holds z = π(x(u, t)) ∈ V and hence z − a ≥ ∆.Therefore, (58) holds provided that

∆ ≥ 10

9

[1 +

1

5η+

√1 + (10η)2

10η

]σ. (59)

In the case η ≤ 1/10, there holds

10

9

[1 +

1

5η+

√1 + (10η)2

10η

]σ ≤ 10

9

[1 + 2 +

√2]σ < 5σ,

and in the case η > 1/10, we estimate

10

9

[1 +

1

5η+

√1 + (10η)2

10η

]σ ≤

[1

9+

2

9+

√1 + (10η)2

9

]σ

η

≤ 3 +√

2

9

σ

η<

σ

2η.

This shows that inequality (59) holds for

∆ :=σ

2 minη, 1/10.

Therefore P does not intersect Bσ(p1) if z − a ≥ ∆. Analogously, P does notintersect Bσ(p2) if b − z ≥ ∆. Since q1 < w < q2 the hyperplane P separatesthese balls, and therefore there holds P ∈ A. This shows B(u, t) ⊂ A whichproves the claim and the lemma.

Provided the curvature is bounded, the boundary condition from lemma 3.39is satisfied on a short time interval. We use this together with lemma 3.31 toobtain a local bound for higher derivative of curvature.

Theorem 3.40. Let x : S1 × [0, t∗] → Rn be a smooth curve shortening flow.Assume there exist an interval I ⊂ S1 and C0 <∞, such that

sup(u,t)∈I×[0,t∗]

|k(u, t)| ≤ C0.

Then for each l ≥ 1 and ρ0 > 0 there exists a constant Cl, depending only on l,ρ0 and C0 such that

|Dlsk(u, t)| ≤ Cl, (u, t) ∈ Ω, (60)

where Ω := (u, t) ∈ I× [δ0, t∗] : µt([a, u]), µt([u, b]) ≥ ρ0 and δ0 := ρ0/(10C0).

50

Proof. For fixed (u0, t0) ∈ Ω, we define

ρ := min

ρ0,

π

4C0e−

C0ρ010

and δ :=

ρ

10C0.

By definition of Ω, the time interval [t0 − δ, t0] is contained in [0, t∗], and theinterval J := [a′, b′] ⊂ S1 defined by

µt0([a′, u0]) = ρ = µt0([u0, b′])),

is contained in I. We define J (1) := [a′, u0] and J (2) := [u0, b′].

By lemma (3.3) there holds

|xu(u, t1)| ≥ |xu(u, t2)| ≥ e−C20 (t2−t1)|xu(u, t1)|. (61)

for all u ∈ I and t1, t2 ∈ [0, t∗] with t1 < t2. In particular, for every intervalI ⊂ I and t1, t2 ∈ [t0 − δ, t0] with t1 < t2 there holds

µt1(I) ≥ µt2(I) ≥ e−C20δµt1(I) = e−

C0ρ10 µt1(I). (62)

By this inequality, for J (i), i = 1, 2 it follows∫J(i)

|k(u, t0 − δ)|dµt0−δ ≤ C0µt0−δ(J(i)) ≤ C0e

C0ρ10 µt0(J (i))

= C0eC0ρ10 ρ ≤ π

4e−

C010 (ρ0−ρ) ≤ π

4.

Therefore, the segment x(J, t0 − δ) is the graph of a Lipschitz function

r0 : V0 → L⊥,

where V0 = [c0, d0] is an interval on the line L = span(T (u0, t0 − δ)), and thereholds

|(r0)z| ≤ η1 := tan(π/4) = 1, z ∈ V0.

By a translation, we may assume that x(u0, t0 − δ) = 0 = (0, r0(0, t0 − δ)), andunder this condition there holds

d0,−c0 ≥µt0−δ(J

(i))√1 + η2

1

≥ µt0(J (i))√1 + η2

1

=ρ√2. (63)

By the curvature bound for all t1, t2 ∈ [0, t∗] and u ∈ I there holds

|x(u, t1)− x(u, t2)| ≤ C0|t1 − t2|.

In particular, for t1, t2 ∈ [t0 − δ, t0] and u ∈ I there holds

|x(u, t1)− x(u, t2)| ≤ σ := C0δ =ρ

10, (64)

By lemma 3.39 for each t ∈ [t0− δ, t0] the segment x(J1(t), t) is the graph ofa Lipschitz map

r(·, t) : V1 ⊂ L→ L⊥, (65)

51

where V1 := [c1, d1] := [c0 + ∆, d0 −∆], J1(t) := u ∈ I : πL(x(u, t)) ∈ V1 and∆ := 5σ, such that there holds

|rz(z, t)| ≤ η2 := 10η1 = 10, (z, t) ∈ V1 × [t0 − δ, t0].

By lemma 3.31, all derivatives of r are bounded in V2 × [t0 − δ/2, t0], whereV2 := [c2, d2] := [c1 +ρ/10, d1−ρ/10]. Since we assume r(0, t0−δ) = 0, for eachm ≥ 1 by lemma 3.31 there exists a constant Cm only depends on m, C0, ρ, δsuch that

|r|m;V2×[t0−δ/2,t0] ≤ Cm. (66)

Since ρ and δ only depend on C0 and ρ0, the constant Cm in fact only dependson m, C0 and ρ0.

Since for a graph flow there holds k = (0, rt)⊥ and the arc-length derivative

is given by Ds = (1 + |rz|2)−1/2Dz, we have

|Dlsk| ≤

∣∣∣∣∣∣(

1√1 + |rz|2

Dz

)l((0, rt)−

〈rz, rt〉1 + |rz|2

(1, rz)

)∣∣∣∣∣∣ .By (66), for every l ≥ 1 there exists a constant Cl depending only on l, C0 andρ0 such that there holds

|Dlsk(u, t)| ≤ Cl, (67)

for all t ∈ [t0 − δ/2, t0] and u ∈ J2(t), where J2(t) := u ∈ I : πL(x(u, t)) ∈ V2.It remains to prove u0 ∈ J2(t). We have to show πL(x(u0, t0)) ∈ V2 = [c2, d2].

By (64) and since we assume that x(u0, t0 − δ) = 0 there holds

πL(x(u0, t0)) ≤ πL(x(u0, t0 − δ)) + |x(u0, t0 − δ)− x(u0, t0)| ≤ σ =ρ

10.

On the other hand by (63) and (64) we have

d2 = d1 −ρ

10= d0 − 5σ − ρ

10≥ ρ√

2− 5

ρ

10− ρ

10≥ ρ

10.

This proves πL(x(u, t)) ≤ d2 and c2 ≤ πL(x(u, t)) follows analogously.

Theorem 3.40 implies the following theorem, that involves the space-timedistance instead of the intrinsic distance. A similar result for immersed hyper-surfaces, is proven by Ecker and Huisken in [20].

Theorem 3.41. Let x : S1 × [0, t∗] → Rn be a smooth curve shortening flow,t0 ∈ (0, t∗] and r ∈ (0,

√t0]. If

supx−1(Pr(x0,t0))

|k| ≤ C0

r,

then there exist Cl <∞ depending only on l and C0, such that

supx−1(Pr/2(x0,t0))

∣∣Dlsk∣∣ ≤ Cl

rl+1,

where Pr(x0, t0) := Br(x0)× [t0 − r2, t0].

52

Proof. By translation and scaling, we may assume that x0 = 0 and r = 1.Fix t1 ∈ [t0 − 1/4, t0] and u1 such that x(u1, t1) ∈ B1/2. We define the timeinterval U := [t1−δ1, t1], where δ1 := min1/4, 1/(4C0). Note that there holdsU ⊂ [t0 − 1, t0]. By the curvature bound, for all t ∈ U there holds

x(u1, t) ∈ B1/4(x(u1, t1)) ⊂ B3/4.

Therefore, the interval I := [a, b] defined by

µt1−δ1([a, u1]) = 1/4 = µt1−δ1([u1, b]),

satisfies x(I, t) ⊂ B1 for all t ∈ U . By theorem 3.40 with ρ0 = (1/8)e−C0/4, foreach l ≥ 1 there exists a constant Cl depending only on l and C0 such that

|Dlsk(u1, t1)| ≤ Cl.

Our next goal is to prove a global version of lemma 3.39. For this reason,we have to study locally Lipschitz graph-like curves a little further.

A circle minimizes the length functional over closed curves x ∈ C2(S1) undera curvature bound and by Fenchel’s inequality there holds

µ(S1) ≥ 2π

sup |k|.

We expect that a regular polygon minimizes the length of closed curves undera Lipschitz ε−graph estimate. However there holds the following correspondinginequality for locally Lipschitz graph-like curves.

Lemma 3.42. Let x : S1 → Rn be a Lipschitz ε−graph. Then the length of xis bounded below by

µ(S1) ≥ πε

4 tan−1(Lipε(x)).

Proof. By the density of C∞(S1,Rn) in C0(S1,R) and W 1,1(S1,Rn), we mayassume that x : S1 → Rn is smooth.

We choose N :=⌈2µ(S1)/ε

⌉ordered points ui ∈ S1, 1 ≤ i ≤ N such that

all intervals Ii = [ui, ui+1], where we use the convention uN+1 = u1, have thesame initial length σ := µ(Ii) ≤ ε/2. By definition of N there holds(

2

ε+

1

µ(S1)

)−1

≤ σ ≤ ε

2.

We define the polygon curve y : S1 → Rn with vertices xi := y(ui) := x(ui)

y(u) := xi + vi(πvi(x(u)− πvi(xi)),

where vi := (xi+1 − xi)/|xi+1 − xi|. The angle between two edges is boundedby ∠(vi, vi+1) ≤ 2β, where β := tan−1(Lipε(x)).

Now we approximate x by a smooth curve z : S1 → Rn by replacing thevertices by smooth segments with absolute curvature less than 2β. For thiscurve, by Fenchel’s inequality and construction there holds

2π ≤∫S1

|k(z)|dµ ≤ 2βN.

53

Together with the estimate above this implies

µ(S1) = Nσ ≥ πσ

β≥ π

β

(2

ε+

1

µ(S1)

)−1

which leads to the inequality

µ(S1) ≥ πε

2β

(1− β

π

)≥ πε

4β.

For further application we need a smooth approximation for locally Lipschitzgraph-like curves with small Lipschitz constant.

Lemma 3.43. Let x : S1 → Rn be a Lipschitz curve with Lipε(x) ≤ 1/10.Then there exist L ∈ L(x)[10/11, 1] and a smooth map z : S1

L → Rn, whereS1L := (L/2π)S1, and a monotone Lipschitz map φ : S1 → S1

L such that forx(u) := x(φ(u)) and all u ∈ S1

L there hold

|zu(u)| = 1, |kz(u)| ≤ 4β, |kzu(u)| ≤ 8β,

|x(u)− z(u)| ≤ βε,1 ≤ |xu(u)| ≤ 11/10,

∠(T x(u), T z(u)) ≤ 2β,

where β := tan−1(Lipε(x)), T x and T z denotes the tangent vectors of x and zrespectively and kz is the curvature of z.

Proof. We the choice of φ, we may assume that x is parametrized by constantspeed |xu(u)| = µ(S1)/(2π) =: λ. We start with the polygon curve y constructedin the proof of lemma 3.42. By the Lipschitz bound of x for u ∈ Ii there holds

∠(T x(u), T y(u)) = ∠(T x(u), vi) ≤ β,

and since we assume that x is parametrized by constant speed there holds

cos(β)λ ≤ |yu(u)| ≤ λ,

|x(u)− y(u)| ≤ tan(β)ε

4.

Since by assumption Lipε(x) ≤ 1/10, lemma 3.42 implies N ≥ 30 and thereforewe have

µ(Ii) ∈ (ε/2)[29/30, 1],

|xi − xi+1| ∈ (ε/2)[9/10, 1].

This lower bound of the length of edges from y allows us to approximate y witha better curve than in lemma 3.42. We can replacing the vertices by smoothsegments to construct a smooth curve z : S1 → Rn such that for all u ∈ S1

there hold

|y(u)− z(u)| ≤ tan(β)ε/4,

cos(β)|yu(u)| ≤ |zu(u)| ≤ |yu(u)|∠(T y(u), T z(u)) ≤ β,

|kz(u)| ≤ 4β,

|kzs(u)| ≤ 8β,

54

where s denotes the arc-length parameter of z. Note that since Lip(0) ≤ 1/10there holds β ≤ Lip(0) ≤ 1.01β. Together with the estimates between x and y,this implies

‖x− z‖C0 ≤ tan(β)ε

2≤ βε,

and for u ∈ Ii

∠(T (u, 0), T z(u)) ≤ ∠(T (u, 0), vi) + ∠(vi, Tz(u)) ≤ 2β.

Finally, we re-parametrize by φ : S1 → S1L, where L := L(z), such that z(u) :=

z(φ(u)) satisfies |zu| = 1. By the estimate

cos2(β)|xu| ≤ |zu| ≤ |xu|,

the map x(u) := x(φ(u)) satisfies |xu| ∈ [1, cos−2(β)] ⊂ [1, 11/10].

Now we are ready to prove a global version of lemma 3.39. The followinglemma is from Altschuler and Grayson, and is an improved version of a lemmafrom Angenent.

Lemma 3.44. (Altschuler-Grayson [8], Lemma 5.4) Let x : S1 × [0, T∗)→ Rnbe a smooth curve shortening flow. Given a positive ε < 10−4 and t0 ∈ [0, T∗)and assume αε(x(·, t0)) ≤ 1/100. Then there holds αε(x(·, t)) ≤ 1/10 for alltime t ∈ [t0, t0 + ε3].

In Altschuler and Grayson’s terminology the lemma shows that an ε−veryflat curve stays ε−flat for a short time. Their proof uses Angenent’s intersectionnumber theory as in lemma 3.39.

We give an alternative lemma, in the spirit of Lipschitz ε−graphs, with adifferent proof.

Lemma 3.45. Let x : S1× [0, T∗)→ Rn be a smooth curve shortening flow. Letε > 0, t0 ∈ [0, T ∗) and assume that Lipε(x(·, t0)) ≤ 1/100. Then for all timest ∈ [t0, t0 + ε2] ∩ [t0, T∗) there holds

Lipε(x(·, t)) ≤ 7Lipε(x(·, t0)) + 12

√Lipε(x(·, t0))(t− t0)

ε2.

Remark. In particular, for Lipε(x(·, t0)) ≤ 10−6 and t ∈ [t0, t0 + ε2] there holdsLipε(x(·, t)) ≤ 1/10.

Proof. We reduce the problem to the case t0 = 0 and ε = 1. This follows by theparabolic rescaling

x(u, t) := ε−1x(u, ε2t+ t0),

for (u, t) ∈ S1 × [0, ε−2(T∗ − t0)). Note that the Lipschitz estimate of a graphis scale invariant and therefore there holds Lipε(x(·, t)) ≤ c if and only ifLip1(x(·, (t− t0)/ε2)) ≤ c.

Hence we have to show that Lip(t) := Lip1(x(·, t)) satisfies

Lip(t) ≤ 7Lip(0) + 12√Lip(0)t

for all t ∈ [0, 1] ∩ [0, ε−2(T∗ − t0)).

55

We define the angleβ := tan−1(Lip(0)).

Note that since Lip(0) ≤ 1/100 there holds β ≤ Lip(0) ≤ 1.001β.Since the conclusion of the lemma does not depend on the parametrization

of x(·, 0), by lemma 3.43, we may assume that x(·, 0) : S1L → Rn and there exists

a smooth map z : S1L → Rn such that for all u ∈ S1

L := (L/2π)S1 there holds

|zu(u)| = 1, |kz(u)| ≤ 4β, |kzu(u)| ≤ 8β, (68)

|x(u, 0)− z(u)| ≤ β, (69)

1 ≤ |xu(u, 0)| ≤ 11/10, (70)

∠(T (u, 0), T z(u)) ≤ 2β, (71)

where T (u, t) := xu(u, t)/|xu(u, t)|, T z(u) := zu(u), kz is the curvature of z.Next we construct an appropriate orthonormal frame of the normal bundle

of z. Therefor, we consider the covariant derivative D⊥∂/∂uX := (∂X/∂u)⊥,

defined on sections X on the normal bundle (Tz)⊥. By the parallel transportof a chosen orthonormal frame eα ∈ (Tu0

z)⊥ at a point u0 along z, we obtain

eα(u) ∈ (Tuz)⊥, 0 ≤ u ≤ L,

that is parallel D⊥∂/∂ueα = 0. We define A ∈ SO(n − 1) by eα(L) = Aαβ eβ(0).

There exists a basis transformation A = StBS such that S ∈ O(n − 1) andB = B(θ1, ..., θN ), θi ∈ (−π, π] is the block diagonal matrix with elements T (θi)of the form

T (θ) :=

(cos(θ) − sin(θ)sin(θ) cos(θ)

)and an additional one if n− 1 is odd, where N := d(n− 1)/2e, i.e. the smallestinteger greater than or equal to (n− 1)/2. We define

B(u) := B

(θ1

Lu, ...,

θNLu

), A(u) := StB(u)S

andeα(u) := B(u)αβ eβ(u).

By definition eα is a global frame on z. By the curvature estimate (68) andFenchel’s inequality there holds π/L ≤ 2β and we conclude∣∣∣D⊥∂/∂ueα(u)

∣∣∣ =

∣∣∣∣( d

duAαβ(u)

)eβ(u)

∣∣∣∣ ≤ max1≤i≤N

|θi|L≤ π

L≤ 2β,

∣∣∣∣ dduD⊥∂/∂ueα∣∣∣∣ =

∣∣∣∣∣(d2

du2Aαβ

)eβ +

(d

duAαβ

)(d

dueβ

)⊥∣∣∣∣∣ ≤ (πL)2

≤ 4β2,

and ∣∣∣∣deαdu∣∣∣∣ =

∣∣∣D⊥∂/∂ueα − 〈kz, eα〉T z∣∣∣ ≤ (π/L) + |kz| ≤ 6β.

With this orthonormal frame eα we define the map

ψ : S1L ×Bn−1

ρ → Rn, (u, y) 7→ ψ(u, y) := z(u) + yαeα(u),

56

where ρ := 1/(8β). Note that ψ is not necessarily injective, but the derivative

Dψ(u, y) =

(T z + yα

deαdu

, e1, ..., en−1

),

satisfies|Dψ − (T z, e1, ..., en−1)| ≤ 6β|y| ≤ 6/8,

and hence ψ is an immersion. For a fixed u∗ ∈ S1L, we define the restriction

ψu∗ : [u∗ − ρ, u∗ + ρ]×Bn−1ρ → Rn, ψ(u, y) := z(u) + yαeα(u).

By integration it follows that φu∗ is a diffeomorphism onto is image. Let φ−1u∗ :

Im(φu∗)→ [u∗− ρ, u∗+ ρ]×Bn−1ρ be the inverse map. Note that for φ−1

u∗ (x) =(w(x), y(x)) the map π(x) := z(w(x)) is the unique nearest point projection ontoz([u∗−ρ, u∗+ρ]). For a later need we estimate the derivatives for |y| ≤ R := 3/2

|Duψ| ∈ [1− 6β|y|, 1 + 6β|y|] ⊂ [9/10, 11/10],

|D2uψ| =

∣∣∣Du

(T z + yαD⊥∂/∂ueα − y

α 〈kz, eα〉T z)∣∣∣

≤ |kz|+ |y|(∣∣∣∣ dduD⊥∂/∂ueα

∣∣∣∣+ |kzu|+ |kz|∣∣∣∣deαdu

∣∣∣∣+ |kz|2)

≤ 4β +3

2

(4β2 + 8β + 4β6β + (4β)2

)≤ 17β.

Claim. There exist a smooth lifting

x = (w, y) : S1L × [0, 1]→ S1

L ×Bn−1ρ ,

such that ψ(w(u, t), y(u, t)) = x(u, t).Proof. For t = 0 we define

x(u, 0) := ψ−1u (x(u, 0)).

By (69) we have |x(u, 0)− z(u)| ≤ β ≤ 1/100 and

|x(u, 0)−z(u+1/4)| ≥ |z(u)−z(u+1/4))|−|x(u, 0)−z(u)| ≥ cos(β)/4−β > 1/10,

and analogously for u− 1/4. Hence there holds |w(u, 0)− u| ≤ 1/4.We can extend x(·, 0) smoothly as long as x(S1

L, t) ⊂⊂ S1L × Bn−1

ρ , i.e. aslong as |y(u, t)| < R.

We claim that there holds |y(u, t)| < β +√

2t. Let g := φ∗gRn be the pullback metric on Zρ := S1

L ×Bn−1ρ . Then x is a smooth curve shortening flow in

(Zρ, g) and we can use shrinking spheres ψ−1u (∂Br(t)(p)) with r(t) =

√r20 − 2t,

p−z(u) ⊥ T z(u) and |p−z(u)| = β+r0 as barriers. Since distg((u, y1), (u, y2)) =distRn(y1, y2) it follows that

|y(u, t)| ≤ β + r0 − r(t) ≤ β +√

2t0,

holds for all 0 ≤ t ≤ t0 := r20/2.

Since |w(u, 0)− u| ≤ 1/4, by the curvature bound of z there holds

∠(T (u, 0), T z(w(u, 0))) ≤ ∠(T (u, 0), T z(u)) + ∠(T z(u), T z(w(u, 0)))

≤ 3β. (72)

57

Let t1 ∈ [0, 1] be the maximal time such that

∠(T (u, t), T z(w(u, t))) ≤ 100β, (73)

holds for all (u, t) ∈ S1L × [0, t1].

Claim. For all (u, t) ∈ S1L × [0, t1] there holds ws(u, t) ∈ [1/2, 2], where

s = s(u, t) denotes the arc-length parameter of x(·, t).Proof. By differentiation of

x(u, t) = ψ(w(u, t), y(u, t)) = z(w(u, t)) + yα(u, t)eα(w(u, t)),

with respect to s we get

T (u, t) = T zws + yαs eα(w) + yα(D⊥∂/∂ueα − 〈kz, eα〉T z)ws.

The scalar product with T z leads to the equation

〈T (u, t), T z〉 = ws(1− yα 〈kz, eα〉).

which implies together with |y| ≤ R and the assumption (73)

ws ≤1

1− |y||kz|≤ 1

1− 6β≤ 2,

ws ≥cos(100β)

1 + |y||kz|≥ cos(100β)

1 + 6β≥ 1/2.

We define the vector field

V (u, t) := T z(w(u, t)).

Locally in ψ([u∗ − ρ, u∗ + ρ] × Bn−1R ), for some fixed u∗ ∈ S1

L, we can extendthis vector field by V (ψ(u, y)) := T z(u). For (u, t) ∈ [u∗ − ρ, u∗ + ρ] × Bn−1

R

there holds V (u, t) = V (x(u, t)) and by the chain rule we have

DV (Duψ) = DuTz = kz.

Since |Duψ| ∈ [9/10, 11/10] the curvature bound (68) implies |DV | ≤ 5β. Forthe second derivative we have

D2V (DuψDuψ) +DV (D2uψ) = D2T z(u) = kzu,

and we conclude

|D2V | ≤ (10/9)2(|kzu|+ |DV | |D2

uψ|)≤ (10/9)2(8β + 5β · 17β) ≤ 11β.

The quantity f(u, t) := 〈T (u, t), V (u, t)〉 satisfies the evolution equation

ft = fss − 2⟨k,DV (T )

⟩−⟨T,D2V (T, T )

⟩+ |k|2f. (74)

By the estimates of |DV | and |D2V | there holds

ft ≥ fss − 10β|k| − 11β + |k|2f.

58

Provided that there exists t2 ∈ (0, T∗) such that

f(u, t) ≥ 5β holds for all t ∈ [0, t2], (75)

in the time interval [0, t2] we have the estimate

0 ≤(|k| − 5β

f

)2

≤ |k|2 − 10|k|βf

+25β2

f2≤ |k|2 − 10|k|β

f+

5β

f,

and this impliesft ≥ fss − 16β.

By (72) there holds f(u, 0) ≥ cos(3β) and hence the maximum principle implies

maxu∈S1

L

f(u, t) ≥ cos(3β)− 16βt,

for all t ∈ [0, t1] ∩ [0, t2]. A posteriori we get the estimate (75) for

t2 =cos(3β)

16β− 5

16≥ 1.

and the assumption (73) holds for

t1 =cos(3β)− cos(100β)

16β≥ 1.

Let I = [u1, u2] ⊂ S1L be a closed interval and t ∈ [0, 1] such that µt(I) = 1.

Then for w1 := w(u1, t) and w2 := w(u2, t) there holds µz([w1, w2]) ≤ 1. Forv = T z((w1 + w2)/2) and u ∈ I we have

∠(T (u, t), v) ≤ ∠(T (u, t), T z(w(u, t))) +∠(T z(w(u, t)), v) ≤ cos−1(f(u, t)) + 2β.

Hence the segment x(I, t) is the graph of a Lipschitz function r(·, t) : [a, b] ⊂L→ L⊥, where L := span(v) and there holds

Lip(r(·, t)) ≤ tan(cos−1(f(u, t)) + 2β)

≤ tan(cos−1(cos(3β)− 16βt) + 2β).

With the estimates cos(z) ≤ 1 − (11/24)z2 and z ≤ tan(z) ≤ (4/π)z for z ∈[0, π/4], this implies

Lip(r(·, t)) ≤ 7Lip(0) + 12√Lip(0)t.

The lemma above together with the local estimate from lemma 3.30 impliesthe following curvature bound. A similar result for the curve shortening flow ina Riemannian manifold which evolves under the Ricci flow, is proven by Morganand Tian, see [46] lemma 18.86.

Lemma 3.46. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Assume that the initial curve satisfies Lipε(x(·, 0)) ≤ 10−6. Then T∗ ≥ ε2 andthere exists C <∞ depending on nothing such that

supu∈S1

|k(u, t)|2 ≤ C

t,

holds for all t ∈ (0, ε2].

59

In view of corollary 3.38 this lemma somehow extends theorem 3.23 in thelimit p to one.

Proof. By lemma 3.45 for t ∈ [0, ε2] ∩ [0, T∗) we have the bound

Lipε(x(·, t)) ≤ 1/10.

Hence we can use theorem 3.30 locally with δ =√

9/10 ε to conclude

supu∈S1

|k(u, t)|2 ≤ 1

t+

δ2

(δ/2)2(δ/2)2=

1

t+

16 · 10

9ε2≤ 19

t,

for all times t ∈ (0, ε2] ∩ (0, T∗). By corollary 3.8 we conclude T∗ ≥ ε2.

Now we can extend the short time existence result from theorem 3.24 toinitial Lipschitz ε−graphs with a small Lipschitz constant.

Theorem 3.47. Let x0 ∈ C0,1(S1,Rn) be a Lipschitz ε−graph. Assume thereholds Lipε(x0) ≤ 10−7 and there exists δ > 0 such that |(x0)u(u)| ≥ δ holdsfor a.e. u ∈ S1. Then there exists a continuous map x : S1 × [0, t∗] → Rn,where t∗ := ε2/2, and a non-decreasing, degree one Lipschitz map φ such thatthe restriction to S1 × (0, t∗] is a smooth curve shortening flow and satisfyingthe initial condition x(φ(u), 0) = x0(u), u ∈ S1.

Proof. We approximate the initial Lipschitz curve by smooth curves x(j)0 ∈

C∞(S1,Rn), such that Lipε(x(j)0 ) ≤ 10−6 for all j ≥ 1 and x

(j)0 → x0 in C0(S1).

Since |(x0)u(u)| ≥ δ for a.e. u ∈ S1, we may assume that x(j)0 are immersed.

Therefore, by short time existence, there exist smooth curve shortening flows

x(j) : S1 × [0, Tj) → Rn with initial conditions x(j)(·, 0) = x(j)0 . Lemma 3.46

implies the uniform estimate

supj≥1

supu∈S1

|k(j)(u, t)|2 ≤ C

t,

for all t ∈ (0, ε2]. In particular, this implies the lower bound Tj ≥ t∗ := ε2.Now, we can argue exactly as in the proof of theorem 3.24.

Finally we combine the estimate from lemma 3.46 with corollary 3.38.

Lemma 3.48. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flowand Kp(t) := ‖k(x(·, t))‖Lp . Then there exists a constant C <∞ such that for1 < p <∞ there holds

supu∈S1

|k(u, t)|2 ≤ C

t, (76)

for all t ∈ (0, ε2], where ε := 10−6pp−1Kp(0)−

pp−1 .

Proof. By corollary 3.38 for Kp(0)ε(p−1)/p < π there holds

Lipε(x(·, 0)) ≤ tan

(Kp(0)ε

p−1p

2

).

60

Since tan(α) ≤ 2α for α ∈ [0, π/4], with the choice ε := 10−6pp−1Kp(0)−

pp−1 there

holds Lipε(x(·, 0)) ≤ 10−6. Therefore, by lemma 3.46 there exists a C < ∞such that

supu∈S1

|k(u, t)|2 ≤ C

t,

holds for t ∈ (0, ε2].

3.8 Angenent’s Regularity Criterion

In this section we generalize two theorems, which are proven by Angenent [5, 6]for a flow of curves on a surface. The first theorem gives a criterion whensingularities can be formed. In the second theorem this criterion is used tobound the number of singular points that can arise at the first singular time.

Theorem 3.49. Let x : S1× [0, T∗)→ Rn be a maximal smooth solution of thecurve shortening flow. Then for any ε > 0 there holds

limtT∗

supI⊂S1

µt(I)≤ε

∫I

|k| dµt ≥ π. (77)

Remarks. In the case of curves on a surface, Angenent ([5], Theorem 9.1) provedthe stronger statement, that if T∗ <∞ then for any ε > 0 there holds

limtT∗

supI⊂S1

µt(I)≤ε

∣∣∣∣∫I

ksc dµt

∣∣∣∣ ≥ π, (78)

where k = kscν. Now, we continue the discussion about the co-dimension, whichwe started on page 44. The quantity on the left hand side of (78) is equal tolimtT∗ αε(x).

For fixed ε > 0, the negation of criterion (77) implies by lemma 3.37 thatthere exists a sequence τj → T∗ such that each curve x(·, τj) is a Lipschitzε−graph with a bound on Lipε(x(·, τj)) that is independent of j ≥ 1. This isthe essential property that is used in the proof of theorem 3.49. Therefore, wecould use proposition 3.35 or the definition of Lipε(x) instead of lemma 3.37 toprove that under the condition of theorem 3.49 for all ε > 0 there holds

limtT∗

αε(x(·, t)) ≥ cn := 2 sin−1

(√n

2(n− 1)

), (79)

as well aslimtT∗

Lipε(x(·, t)) =∞. (80)

So the direct analogue (79) to Angenent’s criterion (78) holds in higher co-dimension only with a constant cn ∈ (π/2, π).

Proof. Let ε > 0. We assume by contradiction that

limtT∗

supI⊂S1

µt(I)≤ε

∫I

|k| dµt ≤ β < π. (81)

61

Since T∗ is maximal, by corollary 3.8 there exists (ui, ti) ∈ S1× [0, T∗) such thatti T∗ and

|k(u, t)| ≤ |k(ui, ti)| → ∞, (i→∞),

for all u ∈ S1 and t ∈ [0, ti]. Since S1 is compact, by taking a subsequence, wemay assume that ui converges to some u ∈ S1.

Recall that by lemma 3.21, the Radon measure µt converges to some Radonmeasure µ∗ as t → T∗. We take the symmetric neighborhood J := (a, b) ⊂ S1

of u such thatµ∗((a, u)) = ε/3 = µ∗((u, b)). (82)

By (81) there exists a sequence τj → T∗ such that there holds∫J

|k| dµτj ≤ β :=β + π

2< π, (83)

for all j ≥ 1.Inequality (83) for some fixed j together Fenchel’s inequality∫

S1

|k| dµτj ≥ 2π,

implies that the interval J is a proper subsets of S1.Moreover, (83) together with lemma 3.37 implies that for each j ≥ 1 the

segment x(J, τj) is the graph of a Lipschitz function

rj : Vj → L⊥j ,

where Vj = (cj , dj) is an interval on a line Lj . Moreover, the Lipschitz constantis uniformly bounded by

|rjz(z)| ≤ η1 := tan(β/2), (84)

for all j ≥ 1 and z ∈ Vj .By lemma 3.21, there exists a Lipschitz map x∗ : S1 → Rn such that x(·, t)

converges uniformly to x∗ as t → T∗. Thus there exists j0 ≥ 1 such that fort0 := τj0 there holds

|x(·, t)− x∗| ≤ σ :=ε

60√

1 + η21

, (85)

for all t ∈ [t0, T∗). Moreover, by (83) and the monotone convergence of theRadon measures µt µ∗, for j0 large enough there holds

ε/3 ≤ µt((a, u)), µt((u, b)) ≤ ε/2, (86)

for all t ∈ [t0, T∗).By (84), (85) and (86), lemma 3.39 implies that for each t ∈ [t0, T∗) the

segment x(J(t), t) is the graph of a Lipschitz map

r(·, t) : V ⊂ L→ L⊥,

where V := [c, d] := [cj0 + ∆, dj0 − ∆] is an interval on the line L := Lj0 ,∆ := σ/(2 minη1, 1/10), and J(t) := u ∈ J : πL(x(u, t)) ∈ V . Moreoverthere holds

|rz(z, t)| ≤ η2 := 10η1,

62

for all (z, t) ∈ V × [t0, T∗). Note that we may assume that η1 ≥ 1/10 and hencethere holds ∆ = 5σ. By (85) and (86) the length of the interval V is boundedbelow by

|V | ≥ µt0(J)√1 + η2

1

− 10σ ≥(

2

3− 1

6

)ε√

1 + η21

=ε

2√

1 + η21

.

Since (ui, ti) → (u, T∗), by taking the tail of the sequence, we may assumethat (ui, ti) ∈ J×[t0, T∗). We rescale the flow by λi := |k(ui, ti)|−1 about (ui, ti)

xi(u, t) := λ−1i

(x(ui + u, ti + λ2

i t)− x(ui, ti)), (87)

for t ∈ Ui := (t0 − ti, T∗ − ti)/λ2i and u ∈ Ji(t) := (ai(t), bi(t)) := J(t)− ui.

For each t ∈ Ui the rescaled segments xi(Ji(t), t) is the graph of the rescaledLipschitz map ri(·, t) : Vi → L⊥

ri(z, t) =: λ−1i

(r(zi + λiz, ti + λ2

i t)− x(ui, ti)),

where z ∈ Vi = [ci, di] = λ−1i (V − zi), zi := πLx(ui, ti). Note that the Lipschitz

constant does not change under the rescaling and hence there holds

|riz(z, t)| ≤ η2, (88)

for all i ≥ 1, t ∈ Ui and z ∈ Vi.By (86) for i large enough there holds µt0([ui, b]) ≥ ε/4, hence by (85) it

follows

di ≥ λ−1i

(µt0([ui, b])√

1 + η21

− 5σ

)≥ λ−1

i

ε

6√

1 + η21

=: λ−1i ρ,

and analogously ci ≤ −λ−1i ρ. This shows that the domain of the map ri contains

the cylinder

Pi := (−λ−1i ρ, λ−1

i ρ)× (λ−2i (t0 − ti), 0] ⊂ Vi × Ui.

These cylinders Pi are monotonically increasing and converge to the set L ×(−∞, 0]. The maps ri : Pi → L⊥ are graph flows satisfying

rit =rizz

1 + |riz|2.

Note that by definition (87) there holds |ki(z, t)| ≤ 1 for (z, t) ∈ Pi. By (88)and lemma 3.31 all the derivatives of ri are locally uniformly in i ≥ 1 bounded.Since ri(0, 0) = 0 this also implies an uniform C0 bound. More precisely, foreach i0 ≥ 1 and every compact set K ⊂ Pi0 there holds

supi≥i0|ri|m;K ≤ C(K,m, β).

Therefore, the maps ri : Pi → L⊥ sub-converges in C∞loc to a limit flowr : L× (−∞, 0]→ L⊥, which satisfies the graph flow equation

rt =rzz

1 + |rz|2.

63

By differentiation, we obtain that p = rz satisfies the parabolic differentialequation in divergence form

pt =

(pz

1 + |p|2

)z

.

Since p is a bounded solution, we can use Moser’s Harnack inequality [45]for each component f1(z, t) := pα(z, t) − infpα : R × (−∞, 0] ≥ 0 as well asf2(z, t) := suppα : R × (−∞, 0] − pα(z, t) ≥ 0, where α ∈ 1, ..., n. Thereexists a constant C = C(β) such that for t0 + 1 ≤ 0 there holds

sup[−1,1]

fj(·, t0) ≤ C inf[−1,1]

fj(·, t0 + 1), j = 1, 2.

By translation and parabolic scaling this implies

fj(z0, t0) ≤ C infR∈[0,

√−t0]

infz∈[−R,R]

fj(z0 + z, t0 +R2) = C infA(z0,t0)

fj ,

where A(z0, t0) := (z, t) ∈ R × (−∞, 0] : |z − z0|2 ≤ t − t0. For j = 1, 2 andz0 ∈ R it follows

limt0→−∞

fj(z0, t0) ≤ C limt0→−∞

infA(z0,t0)

fj = C infR×(−∞,0]

fj = 0.

By definition of f1, f2 this implies

supR×(−∞,0]

pα − infR×(−∞,0]

pα = f2(z0, t0)− f1(z0, t0)→ 0, (t0 → −∞).

Therefore, p = const. and the limit curve x is a straight line. This is a contradic-tion since by definition (87) |ki(0, 0)| = 1 and therefore by smooth convergencethere holds

|pz(0, 0)| ≥ |rt(0, 0)| ≥ |k(0, 0)| = 1.

Our next goal is to localize the blow up argument from the proof above tobound the number of singular space points.

We define regular and singular points at the first singular time.

Definition 3.50. Let x : S1 × [0, T∗)→ Rn be a maximal smooth curve short-ening flow. We call u0 ∈ S1 a regular point of x, if the continuous extensionx : S1 × [0, T∗] → Rn to the first singular time is smooth at (u0, T∗) and thereholds xu(u0, T∗) 6= 0.

We denote by R∗ ⊂ S1 the set of regular points of x, by S∗ := S1 \ R∗ theset of singular points, and by Σ∗ := x(S∗, T∗) ⊂ Rn the set of singular spacepoints.

First we show that a local curvature bound implies that the flow smooth.

Lemma 3.51. Let x : S1× [0, T∗)→ Rn be a maximal smooth curve shorteningflow and u0 ∈ S1. Assume there exist an open neighborhood I ⊂ S1 of u0 andC0 <∞, such that

sup(u,t)∈I×[0,T∗)

|k(u, t)| ≤ C0.

Then there holds u0 ∈ R∗.

64

Proof. By theorem 3.40 for every l ≥ 1 and ρ0 > 0 there exists a constant Cldepending only on l, C0 and ρ0 such that

|Dlsk| ≤ Cl, (u, t) ∈ Ω,

where Ω = (u, t) ∈ I × [δ0, T∗) : µt([a, u]), µt([u, b]) ≥ ρ0, δ0 = ρ0/(10C0) andI =: (a, b).

We define the intervals I(1) := (a, u0) and I(2) := (u0, b). Since x(·, 0) isan immersion and I is an open neighborhood of u0, this defines the positiveconstant

ρ1 := mini=1,2

µ0(I(i)).

By lemma (3.3) there holds

|xu(u, 0)| ≥ |xu(u, t)| ≥ e−C20T∗ |xu(u, 0)|, (u, t) ∈ I × [0, T∗), (89)

In particular, for i = 1, 2 and t ∈ [0, T∗) there holds

µt(I(i)) ≥ ρ2 := e−C

20T∗ρ1.

With the choice ρ0 = ρ2/2, there holds J × [δ0, T∗) ⊂ Ω, where J := [a′, b′] ⊂ Iis defined by

µt([a′, u0]) = ρ2/2 = µt([u0.b

′])).

We parametrize by arc-length at t = δ0. Since x(·, δ0) is an immersion, themap φ : J → R defined by

φ(u) :=

∫ u

u0

|xu(v, t0)|dv.

is a diffeomorphism onto is image V := φ(J), and the map

x(v, t) := x(φ−1(v), t), (v, t) ∈ V × [0, T∗),

defines a smooth curve shorting flow with |xv(v, δ0)| = 1.By lemma 3.7 for every (m, l) ∈ N2 \ 0 there exists a constant Km,l de-

pending only on (m, l), T∗ and Ci for i = 0, ..., 2m+ l − 1 such that

|Dmt D

lvx(v, t)| ≤ Km,l, (v, t) ∈ V × [δ0, T∗).

Since Ci only depends on i, C0 and ρ0, and ρ0 only depends on T∗ and ρ1,the constant Km,l in fact only depends on (m, l), T∗, C0 and ρ1. Hence x(·, t)smoothly extends to V × [δ0, T∗].

Let x : S1 × [0, T∗] → Rn be the continuous extension to the first singulartime. Since φ : J → V is a diffeomorphism, x(u, t) = x(φ(u), t) is smooth inJ × [δ0, T∗]. By (89) x(·, T∗) : J → Rn is an immersion. In particular, thisproves u0 ∈ R∗ as in definition 3.50.

Note that smoothness of φ and x in J × [δ0, T∗] depends on the smoothnessof x(·, δ0).

Now we are ready to show that the number N of singular points in Rn atthe first singular time is bounded by

N ≤ 1

πlimtT∗

K(t),

65

where K(t) denotes the total absolute curvature

K(t) :=

∫S1

|k(u, t)|dµt(u), 0 ≤ t < T∗.

This is proven by Angenent ([6], Theorem 5.1) for curves on a surface.In theorem 5.34, we will prove that for a weak solution, at each singular

time, the number N of singular point in Rn is bounded by

N ≤ 1

climtT∗

K(t),

where c > 0 is a constant depending on nothing.We separately study the first singular time here in order to compare our

result to Angenent’s work. To localize the blow up argument from the proof oftheorem 3.49, the full strength of theorem 5.34 is necessary. That is why, wewill assume here that the set of singular points is already finite. The goal is toimprove the estimate for some c > 0 to the constant c = π. This also applies toweak solutions as long as the singular times do not have an accumulation point.

We have to anticipate some notation that we will use in chapter 5. By lemma3.21, we already know that the limits x∗ = limt→T∗ x(·, t) and µ∗ := limt→T∗ µtexist. At a singularity it is possible that an entire interval I ⊂ S1 collapses toa point, i.e. µ∗(I) = 0. To cover all cases at once, we use the convention that apoint is a closed interval. Hence we will assume that the set singular points R∗in the domain S1 consists of disjoint closed intervals I1, ..., IN , such that eachimage x∗(Ij) is a point pj in Rn. We will actually bound the number of suchsingular intervals in S1.

The next theorem can be compared with Angenent [6], Theorem 5.1.

Theorem 3.52. Let x : S1×[0, T∗)→ Rn be a maximal smooth curve shorteningflow. Let x : S1 × [0, T∗] → Rn denote its continuous extension to the firstsingular time. Assume that the singular set S∗ ⊂ S1 of x consists of disjointclosed intervals I1, ..., IN such that their images at the first singular time arepoints in Rn, i.e. there exist p1, ..., pN ∈ Rn such that x(·, T∗) = pj on Ij. Thenthe number N of such intervals is bounded by

N ≤ 1

πlimtT∗

K(t). (90)

Proof. We separate the closed intervals Ij by open disjoint neighborhoods Jj ⊃Ij . Then there holds

supt∈[0,T∗]

|k(u, t)| <∞, (91)

for all u ∈ ∂Jj , j = 1, ..., N . By lemma 3.51 for each j = 1, ..., N there exists asequence ti T∗ such that

limi→∞

supu∈Jj|k(u, ti)| =∞.

By taking maxima over each Jj , we find sequences ui,j ∈ Jj , j = 1, ..., N suchthat

|k(ui,j , ti)| → ∞, (i→∞), (92)

66

and|k(u, t)| ≤ |k(ui,j , ti)|,

for all u ∈ Jj and t ∈ [0, ti]. By a further subsequence, we may assume thatui,j converges to some uj ∈ Jj as (i → ∞). Clearly there holds uj ∈ Ij , sinceotherwise (92) does not hold.

We define the Radon measure κt on S1 by

κt(φ) :=

∫S1

φ(u)|k(u, t)|dµt(u), 0 ≤ t < T∗,

for φ ∈ C0(S1). By lemma 3.4 κt(S1) = K(t) is non-increasing. By the com-

pactness of Radon measures there exists a sequence τl → T∗ such that the limit

κ∗ := liml→∞

κτl ,

exists and its total measure is bounded by

κ∗(S1) ≤ inf

0≤t<T∗κt(S

1) = limt→T∗

K(t).

Let κ∗ = κcont + κsing be the decomposition of κ∗ with respect to µ∗, i.e.κcont << µ∗ and κsing ⊥ µ∗, where µ∗ := limt→T∗ µt.

Now, we claim αj := κsing(Ij) ≥ π, for all j = 1, ..., N .The theorem follows directly from the claim via the estimate

πN ≤N∑j=1

αj ≤ κ∗(S1) = limtT∗

K(t).

We prove the claim by contradiction and assume there exists j0 = 1, ..., Nsuch that αj0 < π. By the outer regularity of κ∗ there exists a open neighbor-hood J = (a, b) ⊂ Jj0 of Ij0 such that

κ∗(J) ≤ β :=αj0 + π

2< π. (93)

Hence, by taking a subsequence we may assume that

κτl(J) ≤ β :=β + π

2< π, (94)

holds for all l ≥ 1. By the lower semi-continuity of the length functional andsince x(·, T∗) : R∗ → Rn is an immersion and Jj0 \ Ij0 ⊂ R∗, there holds

µ∗((a, uj0)) ≥ L(x(·, T∗)|(a, uj0)) > 0,

and analogously for the interval (u, b). Hence the interval J includes the sym-metric neighborhood J ′ = (a′, b′) defined by

µ∗((a′, uj0)) = ε/3 = µ∗((uj0 , b

′)), (95)

withε := 3 minL(x(·, T∗)|(a, uj0)), L(x(·, T∗)|(uj0 , b)).

Since ui,j0 converges to uj0 ∈ Ij0 , we may assume that ui,j0 ∈ J ′.Now we are in the setting of the proof from theorem 3.49. Equality (95)

corresponds to (82) and the estimate (94) to (83). Hence we can argue as intheorem 3.49 to get a contradiction to (92).

67

3.9 Convex Projection

A compact embedded hypersurface F : Mn → Rn+1 is called convex if thesecond fundamental form satisfies hij ≥ 0, and uniformly convex if hij > 0. Aset U ⊂ Rn is called convex respectively strictly convex, if for every two pointsin the set, the straight line segment between these points is contained in theset U respectively is contained in the interior of the set U . If F : Mn → Rn+1

is (uniformly) convex, then the image F (M) bounds a (strictly) convex set inRn+1.

In [34], Huisken shows that the mean curvature flow preserves convexity andif the initial hypersurface is uniformly convex, then the inequality hij ≥ εHgijis preserves under the flow. Moreover, he proves that the smooth flow existsuntil it shrinks to a round point in finite time. A similar result was provenby Gage and Hamilton [26] for convex curves in R2. In [28], Grayson extendsthis result in the following way: He proves that the curve shortening flow ofembedded curves in R2 becomes convex before the first singular time. Hence,for embedded curves in the plane the smooth flow exists until it shrinks to apoint.

Even though convexity is not defined for curves in Rn, we will extend thisprinciple to higher co-dimension.

We consider the Grassmannian manifold G(k,Rn) of k planes in Rn, withthe usual distance introduced by the operator norm of the projections. i.e. forplanes G1 ∈ G(k,Rn) and G2 ∈ G(l,Rn) the distance is given by

d(G1, G2) := |πG1− πG2

|op := sup|πG1v − πG2

v| : |v| = 1.

If k = l this distance has the geometric description d(G1, G2) = sin(α), whereα is the maximal angle between G1 and G2, i.e. the angle α ∈ [0, π/2] satisfies

cos(α) = inf|v1|=1v1∈G1

sup|v2|=1v2∈G2

〈v1, v2〉.

By adding and subtracting the identity, it follows d(G1, G2) = d(G⊥1 , G⊥2 ). This

shows that the maximal angle between two-planes is equal to the maximal anglebetween their orthogonal complements.

We denote by E(ν) ∈ G(n−1,Rn) the hyperplane with normal ν, and definethe affine hyperplane E(ν, s) := E(ν) + sν for s ∈ R. i.e.

E(ν, s) := x ∈ Rn : 〈x, ν〉 = s.

For a fixed two-plane G ∈ G(2,Rn), we call a hyperplane E(ν, s) normal to Gif its normal satisfies ν ∈ G.

Definition 3.53. A continuous embedding y : S1 → R2 is called embedded(strictly) convex, if there exists a nonempty open (strictly) convex set U suchthat y(S1) = ∂U .

A continuous map y : S1 → R2 is called degenerately convex, if either it is apoint i.e. a constant map y(u) ≡ p, or it traverses an interval twice i.e. up to ahomeomorphism on S1 there holds y(u) = p+ sin(u)v, for some p, v ∈ Rn.

Lemma 3.54. Let y : S1 → R2 be a continuous map. Then the following areequivalent

68

i) #y−1(L) ≤ 2 holds for every line L ⊂ R2,

ii) y is a continuous embedded, strictly convex curve.

Proof. A continuous embedded, strictly convex curve clearly satisfies the condi-tion i). We first observe, that the number of intersections condition i) impliesthat the map y is injective. An injective continuous map y : S1 → R2, by Jordantheorem, divides the plane into an inside and an outside region. If #y−1(L) = 2,the line L splits into three parts. Since the intersection number of all translatedlines is bounded by 2, the middle part of L must lie in the inside region of thecurve.

Corollary 3.55. Let x ∈ C1(S1,Rn) be an immersed curve, and let G ∈G(2,Rn) be a two-plane. Then the projection πGx is a continuous embedded,strictly convex curve if and only if the number of intersections #x−1(E) ≤ 2,for every hyperplane E normal to G, i.e. for all hyperplane E = E(ν, s) withν ∈ G.

Proof. For the number of intersections there holds

#x−1(E) = #(πGx)−1(L),

where E is a hyperplane normal to G and L = πGE is a line in G. Hence, thecorollary follows immediately from the lemma 3.54

Lemma 3.56. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow,and let G ∈ G(2,Rn) be a two-plane. Assume that the projection πGx(·, 0) is acontinuous embedded, strictly convex curve.

Then the projection y(·, t) := πGx(·, t) is an immersed smooth embedded,strictly convex curve, for all time t ∈ [0, T∗). Moreover, the map x(·, t) is asmooth embedding for all times t ∈ [0, T∗).

Proof. By corollary 3.55 and theorem 3.11, it follows immediately that the pro-jection y(·, t) is an injective, strictly convex curve.

We claim that y(·, t) is a smooth immersion for all t ∈ [0, T∗). Otherwisethere exists (u0, t0) ∈ S1 × (0, T∗) such that πG(xu(u0, t0)) = 0. Therefore, thecurve x(·, t0) at q := x(u0, t0) is tangential to the hyperplane

E(ν, q) := p ∈ Rn : 〈p, ν〉 = 〈q, ν〉

for all ν ∈ G with |ν| = 1. By theorem 3.11 the intersection number

iE(ν,q)(t) := #u ∈ S1 : x(u, t) ∈ E(ν, q)

satisfies iE(ν,q)(t) ≤ 1 for all t > t0. Since this holds for all ν ∈ G with|ν| = 1, the map x(·, t) is constant for t > t0. This implies t0 = T∗, which is acontradiction.

It is left to prove that x(·, t) is a smooth embedding. By definition of thesmooth flow it is already a smooth immersion. Since the projection y(·, t) isinjective, the map x(·, t) is injective, and hence a smooth embedding.

Lemma 3.57. Let y ∈ C0(S1,R2) be a continuous embedded convex curve.Then y is a locally Lipschitz graph-like curve.

69

Proof. The map y bounds an open convex set C ⊂ R2. By compactness, wehave the positive number

d0 := infe∈S1

H1(πe(C)) > 0.

Every segment of y with length less than d0 can be written as graph of a convexfunction f : I ⊂ R → R. As a convex function f is locally Lipschitz, thederivative f ′ exists almost everywhere. Moreover, f ′ is monotone, and hencecontinuous in a co-countable set, and it follows that the derivative f ′ existsactually in a co-countable set. In particular, the curve y is a locally Lipschitzgraph-like curve, and in fact a Lipschitz ε−graph for every ε < d0.

Now the goal is to prove following statement: The property that a projectionπGx is an embedded uniformly convex curve is an open condition with respectto the two-plane G in the Grassmannian manifold.

Let x ∈ C1(S1,Rn) be an immersed curve and G a two-plane. The projectiony := πG(x) is an immersed C1 curve if and only if the projection of the tangentvector τ = πGT does not vanish. Hence if y is immersed, by compactness of S1,the positive distance

d1 := dist(T (S1), G⊥) = inf|τ(u)| : u ∈ S1 > 0,

will be attained.Let x ∈ C2(S1,Rn) be an immersed curve and G a two-plane. Assume the

projection y := πG(x) is an immersed, uniformly convex C2 curve. Since y isuniformly convex its curvature satisfies k(y) > 0.

The curvature k(y) of the projected curve can be computed in terms of theprojection of the curvature κ := πGk(x) and the tangent vector τ = πGT

k(y) =κ

|τ |2− 〈κ, τ〉τ|τ |4

.

Hence k(y) > 0 implies |k(x)| ≥ |κ| > 0 and actually there holds

|k(y)|2 =|κ|2

|τ |4− 〈κ, τ〉

2

|τ |6.

By compactness of S1, we have the positive constant

d2 := inf

|k(y)(u)||k(x)(u)|

: u ∈ S1

> 0.

Lemma 3.58. Let x ∈ C1(S1,Rn) be a immersed curve and G ∈ G(2,Rn) atwo-plane.

i) Assume that the projection y := πG(x) is a immersed curve. Then for everytwo-plane G′ ∈ G(2,Rn) with

d(G,G′) ≤ d1

4, (96)

the projection πG′x is also an immersed curve.

70

ii) Assume that the projection y := πG(x) is an embedded, uniformly convexcurve. There exists a positive constant c0 > 0 (depending on nothing) suchthat for every two-plane G′ which satisfies (96) and

d(G,G′) ≤ c0d31d2, (97)

the projection πG′(x) is an embedded uniformly convex curve.

For a fixed curve x ∈ C2(S1,Rn) and a fixed two-plane G, we call a hy-perplane E = E(ν, s) admissible with respect to x over G, if E is normal to atwo-plane G′ with

d(G,G′) < η := mind1/4, c0d

31d2

,

or, equivalently if the projection of the normal ν of E lies in the open region

V := ν ∈ Sn−1 : |πG⊥(ν)| < η.

Proof. Let β be the minimal angle between G⊥ and T (S1). Since y := πG(x)is immersed, by definition of d1 there holds sin(β) = d1. As mentioned above,the maximal angle α between G⊥ and G′⊥ satisfies d(G,G′) = sin(α). Byassumption (96) we have sin(α) ≤ sin(β)/4 which imply α ≤ β/2. Hence theminimal angle between G′⊥ and T (S1) is bounded by β − α ≥ β/2. Thereforethe projection of the curve to G′ is an immersion and we have the estimate

|πG′T | ≥ sin(β/2) ≥ d1/4.

To prove the second statement, let y′ := πG′(x), κ′ = πG′k(x), and τ ′ = πG′T .We have the point-wise estimate

|k(y)− k(y′)| ≤∣∣∣∣ κ|τ |2 − κ′

|τ ′|2

∣∣∣∣+

∣∣∣∣ 〈κ, τ〉τ|τ |4− 〈κ

′, τ ′〉τ ′

|τ ′|4

∣∣∣∣The first term is bounded by

(|τ |+ |τ ′|)|τ − τ ′||κ|+ |τ ||κ− κ′||τ |2|τ ′|2

≤ 32|k(x)|d3

1

d(G,G′),

and the second term is estimated by

|〈κ, τ〉||τ − τ ′|4

|τ |3|τ ′|4+

(|τ ′||κ− κ′|+ |κ||τ − τ ′|)|τ ′|3

+|κ′||τ − τ ′||τ ′|3

≤ 81|k(x)|d3

1

d(G,G′).

Together, we have the estimate

|k(y)− k(y′)| ≤ 113|k(x)|d3

1

d(G,G′).

With the choice c0 = 1/226 it follows |k(y) − k(y′)| ≤ |k(y)|/2. Hence y′ isuniformly convex with the lower bound |k(y′)| ≥ |k(y)|/2 > 0.

71

Theorem 3.59. Let x : S1×[0, T∗)→ Rn be a maximal smooth curve shorteningflow and let G ∈ G(2,Rn) be a two-plane. Assume that the projection πGx(·, 0)is a continuous embedded, uniformly convex curve. Then the limit curve x∗ =limt→T∗ x(·, t) is degenerately convex. i.e. x∗ is either constant or an interval.

Proof. By lemma 3.21 the curve x(·, t) converges uniformly to a limit curve x∗ ast tends to T∗. By lemmata 3.56 and 3.58 there exists an η > 0, depending only onthe initial curve and its projection, such that the projection y(·, t) := πG′x(·, t)is a smooth embedded, strictly convex curve for every time t ∈ [0, T∗) and alltwo-planes G′ in the set

V1 := G′ ∈ G(2, n) : d(G,G′) < η.

For all G′ ∈ V1 we can write the curve x(·, t) as a graph over y(·, t),

x(u, t) = (y(u, t), f(y(u, t), t)) .

Moreover, in the smaller set

V2 := G′ ∈ G(2, n) : d(G,G′) < η/2,

there holds the Lipschitz bound Lip(f(·, t)) ≤ Lip(f) := cot(η/2). The mainpoint is, that η and hence also Lip(f) are independent of the time t.

As a limit of convex maps, the projection πG′x∗ is either convex or degen-erately convex for all G′ ∈ V2.

Now, we argue by contradiction and assume that the limit curve x∗ is notdegenerately convex. Since V2 is an open subset of the Grassmannian, it is notpossible that for all G′ ∈ V2 the projections πG′x∗ are degenerately convex.

Hence there exists a G′ ∈ V2 such that the projection y∗ = πG′x∗ is acontinuous embedded convex curve. By lemma 3.57, the map y∗ is locallyLipschitz graph-like, i.e. there exists ε > 0 such that Lipε(y∗) <∞.

Therefore, the curve x∗ is a Lipschitz graph over the Lipschitz ε−graphy∗ = πGx∗. Next, we show that this implies that x∗ itself is a Lipschitz ε−graph.If a segment y(I, t) is the graph of a Lipschitz function

r(·, t) : J ⊂ R→ R,

then the segment x(I, t) is the graph of the function

R(·, t) : J → Rn−1, R(z, t) := (r(z, t), f(r(z, t), t)).

Since the length of x(I, t) is larger than the length of y(I, t), this leads to theestimate

Lipε(x(·, t)) ≤√

1 + Lip(f(·, t))2 Lipε(y(·, t)) ≤ Lipε(y(·, t))sin(η/2)

.

In the limit t→ T∗ this implies

Lipε(x∗) ≤Lipε(y∗)

sin(η/2)<∞.

By theorem 3.49, this contradicts the assumption that T∗ is the first singulartime.

72

We additionally prove that if the intersection number is bounded by two forall hyperplanes then the curve lies in a plane and is convex.

Lemma 3.60. Let x : S1 → Rn be a immersed, real analytic curve such that forall hyperplanes E ⊂ Rn there holds either #x−1(E) ≤ 2 or x(S1) ⊂ E. Then

i) there exists a plane G with dim(G) = 2 such that x(S1) ⊂ G,

ii) x : S1 → G is a continuous embedded strictly convex curve.

Remark. For j ≥ 1 the condition that for every hyperplane E either #x−1(E) ≤2j or x(S1) ⊂ E holds, still implies that x(S1) is contained in a 2j−dimensionalplane.

Proof. We use a local arc-length parametrization x : I → Rn with x(0) = 0 and|∂ux(0)| = 1,

x(u) =∑j≥1

uj

j!∂jux(0).

Let i ≥ 2 be the first integer such that ∂iux(0) /∈ span(∂ux(0)). If there is no suchi ≥ 2 the curve is a straight line, which contradicts the assumption that the curveis closed. We claim that for all l ≥ i there holds ∂lux(0) ∈ span(∂ux(0), ∂iux(0)).Otherwise e1 = ∂ux(0), e2 = ∂iux(0) and e3 = ∂lux(0) are linearly independent.Locally the curve can be written as

x(u) = e1u+ e2ui

i!+ e3

ul

l!+O(ul + 1).

There exist u1, u2, u3 near u = 0 such that x(u1), x(u2) and x(u3) are linearlyindependent. Let E be the hyperplane spanned by x(u1), x(u2) and the or-thonormal space of span(e1, e2, e3). By construction we have #x−1(P ) ≥ 3 andx(u3) /∈ E which contradicts the assumption. Therefore, the curve is containedin the two-plane G = span(e1, e2).

To proof that x(S1) ⊂ G bounds a strictly convex set, it is enough to showthat the i ≥ 2 from above is an even integer. Otherwise, since

x(u) = e1u+ e2ui

i!+O(ui+1),

there exists u1 < 0 < u2 such that the projections of x(u1) and x(u2) tospan(e1, e2) are linearly dependent. Let E be the hyperplane spanned by x(u1)and the orthonormal complement of span(e1, e2). Since x(u1), x(0), x(u2) ∈ Ethere holds x(S1) ⊂ E. But we already know that the curve lies in the plane G,hence x(S1) ⊂ span(x(u1)). This is a contradiction since a real analytic closedcurve can not lie entirely in line.

3.10 Local Regularity

The maximal area ratio is defined by

Θ(t) := supp∈Rn

supr>0

µt(x−1(Br(p), t))

2r.

The following ε−regularity theorem was conjectured by Tom Ilmanen andthe proof idea is also from him.

73

Theorem 3.61. (Hattenschweiler-Ilmanen) For every D0 > 0, there exist con-stants ε0 > 0 and C0 < ∞, such that for every smooth curve shortening flowx : S1 × [0, T ) → Rn with initial bounded maximal area ratio Θ(0) ≤ D0 thefollowing holds: If for r > 0, t0 ∈ [r2, T∗] and x0 ∈ Rn there holds

r−1

∫ t0

t0−r2

∫Dr(x0,t)

|k|2dµtdt ≤ ε2 ≤ ε20, (98)

where Dr(x0, t) := x(·, t)−1(Br(x0)), then the curvature is bounded

supx−1(Pr/2(x0,t0))

|k(u, t)| ≤ C0ε

r, (99)

in the parabolic cylinder Pr/2(x0, t0) := Br/2(x0)× [t0 − (r/2)2, t0).

Remarks. The curvature bound (99) together with theorem 3.41 shows that allderivatives of the curvature are bounded in the smaller cylinder Pr/4(x0, t0). Inthis smaller parabolic cylinder the flow smoothly extends to Br/4(x0)× t0.

A related result for embedded hypersurfaces is the following theorem.

Theorem 3.62. (Ecker [17], Theorem 2.1) There exist constants ε0 > 0 andC > 0 such that for any embedded mean curvature flow F : M2 × [0, T ) → R3,any x0 ∈ R3 and r ∈ (0,

√T ) the inequality

sup(T−r2,T )

∫Mt∩Br(x0)

|A|2 dµt ≤ ε0,

implies the estimate

supPr/2(x0,T )

|A|2 ≤ C

r4

∫ T

T−r2

∫Mt∩Br(x0)

|A|2 dµtdt.

Compared to this result our assumption (98) involve the integral instate ofthe supremum over the time interval. An other related results for the meancurvature flow with arbitrary dimension and co-dimension is following theorem

Theorem 3.63. (Ilmanen [39], Theorem 14) There exists ε0 = ε0(n, k) > 0 suchthat if (Mk

t )t∈[0,1) is a mean curvature flow smoothly immersed in B1 ⊂ Rn,and

r−k∫ t

t−r2

∫Mt∩Br(x)

|A|2 dµtdt ≤ ε2 ≤ ε20,

whenever Br(x)× [t−r2, t) ⊂ B1× [0, 1), then Mt∩B1 can be smoothly extendedto B1 × 1 and

|A| ≤ C(n, k)max

1

1− |x|,

1

t1/2

ε

for all (x, t) ∈ B1 × (0, 1].

Note that in our theorem 3.61 the scale invariant estimate (98) only is re-quired on a fixed parabolic cylinder Pr(x0, t0). This assumption is much weakercompared to this theorem, where the estimate is required at every point for allsmall scales. Further result can be found in Han-Sun [33] and Ecker [19], whichare improved versions of theorem 3.63.

74

A main application of all these ε−regularity theorems, is a bound on theHausdorff dimension of the singular set at the first singular time. For a weaksolution an ε−regularity theorem implies a bound of the parabolic Hausdorffdimension of the singular space-time points.

In Deckelnick’s work [12], lemma 3.2 is an ε−regularity theorem for his weaksolutions to the equation xt = xuu/|xu|2. In this lemma || |xu(., t)|−α||L∞(S1),α ∈ (0, 1) plays the roll of

∫|k|2 dµt. This quantity is not geometric and not lo-

cally in space. However, it leads to the same result that the Hausdorff dimensionof the set of singular times is less than 1/2. Since Deckelnick’s quantity includesthe parameter α, he only can prove that the Hausdorff 1/2 + δ measure is finitefor all δ > 0, whereas our regularity theorem will imply that the Hausdorff 1/2measure of the set of singular times is finite.

Our proof is distinguish to all other proofs of the results mention above.By an argument that is based on Angenent’s intersection theory we will reducethe problem to independent components that are Lipschitz graphs. For each ofthis components we will use theorem 3.30 to get a curvature bound. Since theGaussian density ratio of such components is nearly one, it is also possible toapply White’s local regularity theorem [51], or Brakke’s regularity theorem [9]to obtain a curvature estimate.

Proof of Theorem 3.61.

A) By the scale- and translation-invariance of the flow, we may assume thatr = 1, x0 = 0 and t0 = 0. Hence, assumption (98) reduces to∫ 0

−1

∫D1(t)

|k|2dµtdt ≤ ε2, (100)

where Dr(t) := Dr(0, t) := x(·, t)−1(Br(0)).

B) Claim. It is enough to prove the estimate |k| ≤ Cε/r in a very smallparabolic cylinder Pλr(x0, t0) for some 0 < λ < 1/2.Proof. We actually claim that if the condition (98) with ε′0 implies theestimate |k| ≤ C ′ε/r in the cylinder Pλr(x0, t0), then the condition (98)with ε0 := ε′0/

√2 implies the estimate |k| ≤ Cε/r with C := 2

√2C ′ in the

cylinder Pr/2(x0, t0).

For (x′0, t′0) ∈ Pr/2(x0, t0), we define P ′ := Pr/2(x′0, t

′0) ⊂ Pr(x0, t0) =: P .

The condition (98) with ε0 in the cylinder P implies∫P ′|k|2 dµtdt ≤

∫P

|k|2 dµtdt ≤ ε20r = ε′0

2(r/2).

Hence the condition (98) with ε′0 holds in the cylinder P ′, and by assumptionwe conclude

|k(x′0, t′0)| ≤ C ′

r/2

(1

r/2

∫P ′|k|2 dµtdt

)1/2

≤ C

r

(1

r

∫P

|k|2 dµtdt)1/2

.

Since (x′0, t′0) ∈ Pr/2(x0, t0) was arbitrary this proves the claim.

C) Claim. There exists a constant C, depending on noting, such that

Θ(t) ≤ D := CD0, t ∈ [0, T∗).

75

Proof. By proposition D.2 for all p ∈ Rn,r > 0 and t ∈ [0, T∗) there holds

Θ(p, r, t) ≤ C Θ(p, r0, 0) ≤ CD0,

for r0 :=√

6 maxr,√t and C := 32e1/4(3π)1/2. This implies the claim via

Θ(t) := supp∈Rn

supr>0

Θ(p, r, t) ≤ CD0.

D) Claim. We can assume that x(·,−1) is cone like in the sense that∫D1(−1)

|k|2 dµ−1 ≤ ε2, (101)

and ∫D1(−1)

|x⊥|2 dµ−1 ≤ ε2. (102)

Proof. By Huisken’s monotonicity formula, which is stated in theorem D.1,the Gaussian density ratio

ψ(t) :=

∫S1

%(x(u, t), t) dµt(u),

is monotone non-increasing and satisfies

d

dtψ(t) = −

∫S1

%

∣∣∣∣k +x⊥

2(−t)

∣∣∣∣2 dµt.By compactness, we may assume that x(S1,−1) ⊂ BR(0) for some R > 2.Hence, by the assumption that the flow has bounded area ratio, we have

ψ(−1) =1√4π

∫S1

exp

(−|x(u, 0)|2

4

)dµ−1

≤ 1√4π

∫D2

dµ−1 +1√4π

∫DR\D2

4

|x(u, 0)|2dµ−1

≤ µ−1(D2)√4π

+4√4π

∫ R

2

∫|x(u,0)|=r

dH0(u)

r2dr

≤ 2√π

µ−1(D2)

4+

4√π

µ−1(DR)

2R≤ 4D.

We choose the sequence si := −2−i. Therefore, for all δ > 0 there existsj ≤ 4D/δ such that∫ si+1

si

∫S1

%

∣∣∣∣k +x⊥

2(−t)

∣∣∣∣2 dµtdt = ψ(sj)− ψ(sj+1) ≤ δ.

From this inequality and assumption (100), for the interval I := [sj , sj+1]there holds

H1

t ∈ I |

∫D1(t)

%

∣∣∣∣k +x⊥

2(−t)

∣∣∣∣2 dµt ≤ 4δ

|I|

≥ 3|I|

4,

H1

t ∈ I |

∫D1(t)

|k|2dµt ≤4ε2

|I|

≥ 3|I|

4.

76

Therefore, there exists a time t1 ∈ I such that∫D1(t1)

ρ

∣∣∣∣k +x⊥

2(−t1)

∣∣∣∣2 dµt1 ≤ 4δ

|I|∫D1(t1)

|k|2dµt1 ≤4ε2

|I|.

Since the backward heat kernel

%(x, t) :=1√

4π(−t)exp

(− |x|

2

(−4t)

),

for |x| ≤ r :=√−t1 is bounded from below by

%(x, t) ≥ e−1/4

√4πr

=1

Cr,

together with the estimate 2−(4D/δ+1) ≤ 2−(j+1) = |I| ≤ r2 ≤ 2−j andr2 ≤ 2|I|, we have

r

∫Dr(t1)

∣∣∣∣k +x⊥

2(−t1)

∣∣∣∣2 dµt1 ≤ Cδr

∫Dr(t1)

|k|2dµt1 ≤2ε2

r≤ C24D/δε2. (103)

Finally, this leads to the inequality

r−3

∫Dr(t1)

|x⊥|2dµt1 ≤ C(

24D/δε2 + δ). (104)

For any given ε > 0, we first choose δ = ε2/2C and second choose ε0 > 0such that C24D/δε2

0 ≤ ε2/2. Hence both inequalities (103) and (104) areless than ε2. Since both quantities are scale invariant, estimates (101) and(102) hold for the rescaled flow x(u, t) := λx(u, λ−2t) with scaling factor

λ = r−1 ≤ 2CD/ε20 .

E) Let U ⊂ Rn be an open set and x : S1 → Rn be a continuous map. Wecall the restriction y := xbI a component of x in U , if I is a connectedcomponent of the pre-image x−1(U) ⊂ S1. The image of a component willbe denoted by σ = y(I).Claim. Each component of x(·,−1) in B1 is an embedded curve and can bewritten as the graph of a function w : V ⊂ L → L⊥ ∼= Rn−1 over a line L,such that there hold the estimates |w|, |w′| ≤ ∆1 := Cε, where C = 4

π .

Proof. Let y = xbI be a component of x(·,−1) in the ball B1. i.e. I is aconnected component of D1(−1). By estimate (101) and corollary 3.38, forε < π/2 there holds Lip4(y) ≤ tan(ε). We choose ε < π/4, so that thereholds

Lip4(y) = tan (ε) ≤ 4

πε.

Hence each segment of y with length less than 4 is the graph of a functionw with the required Lipschitz bound. If we choose ε small enough such that

77

√1 + Lip4(y)2 ≤ 2, the length of the whole component is less than 4. Hence

the whole component can be written as the graph of a function w with theestimate |w′| ≤ Cε and by a translation the same estimate holds for |w|.

F) Now we are going to show a cone like estimate, which arises from property(102). To this end, we consider a component at time t = −1 and thecorresponding line L, constructed in item (E).Claim. The distance d = dist(L, 0) is either small

d ≤ ∆2, (105)

or bigd ≥ 1−∆2. (106)

where ∆2 = Cε.

Proof. We write x = (y, z) ∈ Rn−1 × R and assume that L is generated bythe vector (0, 1). We can assume that d+ ∆1 < 1 since otherwise inequality(106) holds trivially for ∆2 = ∆1. Hence, we have a lower bound for thelength l := length(σ) of the component

l ≥ 2√

1− (d+ ∆1)2.

The normal part of x with respect to σ satisfies

x⊥ = x− 〈x, τ〉 τ,

where x = (y, z) = (v, 0) + (w, z) and τ = (w′, 1)/√

1 + |w′|2. We claimthat the difference between x⊥ and the normal part of x with respect to L,x⊥L := (y, 0) is small, in the sense that

|x⊥ − x⊥L|2 ≤ C|w′|2 ≤ C∆21. (107)

This follows by the calculation

|x⊥ − x⊥L|2 = |(0, z)− 〈x, τ〉 τ |2

≤

∣∣∣∣∣〈x, τ〉 w′√1 + |w′|2

∣∣∣∣∣2

+

∣∣∣∣∣z − 〈x, τ〉√1 + |w′|2

∣∣∣∣∣2

.

The first term on the right hand side is bounded by |w′|2 ≤ ∆21. For the

second we calculate

〈x, τ〉 =

⟨(v, 0) + (w, z),

(w′, 1)√1 + |w′|2

⟩

=〈v, w′〉√1 + |w′|2

+z√

1 + |w′|2+

〈w,w′〉√1 + |w′|2

The squares of the first term and last term are bounded by ∆21. For the

middle term we have∣∣∣∣∣z − z√1 + |w′|2

∣∣∣∣∣2

=|z|2|w′|2

1 + |w′|2≤ ∆2

1.

78

By the estimate (A+B)2 ≤ 2(A2 +B2) this shows inequality (107).

Note that we can assume that d > ∆1 since otherwise inequality (105) holdstrivially for ∆2 = ∆1. Since |x⊥L| ≥ d−∆1 and σ is cone like, we have

l(d−∆1) ≤∫σ

|x⊥L|

≤∫σ

|x⊥|+∫σ

|x⊥ − x⊥L|

≤ l1/2(∫

σ

|x⊥|2)1/2

+ C∆1l

≤ l1/2∆1 + C∆1l.

We choose ε0 > 0 small enough such that ∆1 = (4/π)ε < 1/4.

Case 1) d ≤ 1/2In this case we have a lower bound on the length

l ≥ 2√

1− (d+ ∆1)2 ≥√

7/2.

Hence the inequality above implies d ≤ C∆1.

Case 2) d ≥ 1/2In this case we have d−∆1 ≥ 1/4 and we conclude

l/4 ≤ l1/2∆1 + C∆1l

Since the length is bounded from above by l ≤ 2√

1 + |∆1|2 ≤ 4 this implies√1− (d+ ∆1)2 ≤ l ≤ C∆1,

and this implies d ≥ 1− C∆1.

G) Claim. There exist a radius r1 ∈ [0.9, 1] and p lines Li with 0 ∈ Li suchthat

x(S1, t) ∩Br1 ⊂p⋃i=1

N∆3(Li), (108)

for all t ∈ [−1, 0], where ∆3 = C√D√ε.

Proof. By item (F), the inclusion (108) holds at time t = −1 for ∆3 =∆1 +∆2 in a ball of radius r0 = 1−∆2. Since the length of such componentsis bounded from below, their number is bounded by p ≤ CD.

We define r1 := r0 − (2 +√D)√ε and fix v ∈ S1 and t1 ∈ [−1, 0] such

that x(v, t1) ∈ Br1 . Then there exist u1 ≤ v ≤ u2 such that the length ofx([u1, u2], t1) satisfies lt1(u1, u2) =

√ε.

First we assume that x((u1, u2), t) ⊂ Br0 for all t ∈ [−1, t1]. Under thisassumption for t2 ∈ [−1, t1] we have

lt2(u1, u2) = lt1(u1, u2) +

∫ t1

t2

∫ u2

u1

|k|2dµtdt ≤√ε+ ε2 ≤ 2

√ε (109)

79

and by Holder inequality there holds

√Dε ≥

∫ t1

t2

∫Dr0 (t)

|k|dµtdt ≥∫ u2

u1

∫ t1

t2

|xt(u, t)|dtdµt1 . (110)

By inequality (110) there exists a u ∈ [u1, u2] such that∫ t1

t2

|xt(u, t)|dt ≤√Dε

lt1(u1, u2)=√Dε. (111)

Since x(v, t1) ∈ Br1 and lt1(u1, u2) =√ε we have x(u, t1) ∈ Br1+

√ε. To-

gether with inequality (111) this implies x(u, t2) ∈ Br1+(1+√D)√ε and by

inequality (109) it follows x((u1, u2), t2) ⊂ Br1+(2+√D)√ε = Br0 . Since

t2 ∈ [−1, t1] was arbitrary this proves x((u1, u2), t) ⊂ Br0 for all t ∈ [−1, t1].

Finally, we have the inequality

d(x(v, t1), L) ≤ d(x(v, t1), x(u, t1)) + d(x(u, t1), x(u,−1)) + d(x(u,−1), L)

≤√ε+√Dε+ (∆1 + ∆2) ≤ C

√D√ε.

The claim follows since we can choose ε0 small enough such that r1 ≥ 0.9.

H) Claim. It is enough to show the theorem for a squeezed flow, i.e. such thatthere exists one line L 3 0 with

x(S1, t) ∩Br1 ⊂ N∆4(L), (112)

for all times t ∈ [−1, 0), where ∆4 = CD3/2√ε.

Proof. Our aim is to collect near neighborhoods N∆3(Li), i = 1, ..., p to-

gether. Let U be the projection to the real projective space Pn(R) of

p⋃i=1

N∆(Li) ∩ ∂Br1 ,

and let Uj , j = 1, ..., q be the connected components of U . We collect allneighborhoods N∆3

(Li) together whose boundaries lie in the same compo-nent Uj . Neighborhoods of different collections have disjoint boundarieson the ball Br1 . Therefore, the flows of the components which lie in suchneighborhoods do not interact inside Br1 . Finally, we replace each collectionby one neighborhood N∆3pj (Lj), where pj denotes the number of neighbor-hoods in the collection from Uj . This proves the claim since pj ≤ p ≤ CD.

I) Claim. Assume that the flow is squeezed according to (112) and consists attime t = −1 of p Lipschitz graphs over the line L with slope less than ∆4.Then there exists a set U ⊃ B1/2∩N∆4

(L) such that for all times t ∈ [−1, 0]the flow x(·, t) has exactly p components yi(u, t) in U . Moreover, at eachtime each component connects the two boundaries of ∂U ∩N∆4(L).

To prove this claim we need the following lemma.

Lemma 3.64. With the same assumption as in the claim, for each ballBr(x0) ⊂ B3/4(0) with r = 1/8 and x0 ∈ L, there exists an (n− 1)-plane Psuch that tan(∠(L,P )) ≥ ∆4 and #u ∈ S1 : x(u, 0) ∈ Br(x0) ∩ P ≥ p.

80

Proof of the claim. We use the lemma for two balls Br(xi) ⊂ B1, which lie intwo complementary hemispheres. More explicitly, we choose x1 = (0, 3/4),x2 = (0,−3/4) and r = 1/8. By the lemma there exist (n− 1)-planes P1, P2

with such that #x(·, 0)−1(Pi ∩ Br(xi) ≥ p and tan(∠(L,Pi)) ≥ ∆4. Sincethe flow is squeezed, by theorem 3.11 these numbers of intersections arenon-increasing. Since at time t = −1 the flow consists of p Lipschitz graphswith the slope less than ∆4, both numbers of intersection points are exactlyp. Therefore, there holds

#u ∈ S1 : x(u, t) ∈ Pi ∩Br(xi) = p,

for all times t ∈ [−1, 0]. Hence in the region N∆4(L) between P1 and P2 theflow consists of p independent components.

Proof of the lemma. Fix x0 ∈ L and r > 0 such that Br(x0) ⊂ B1(0). Byassumption at the time t = −1 we have p Lipschitz graphs over L withmaximal slope by ∆4. Recall that ∆4 = CD3/2

√ε and we will choose ε0

small enough such that ∆4 ≤ 1/10. We define the angle α by tan(α) = ∆4

and the set

A := P ⊂ Rn : P is an (n− 1)-Plane, ∠(L,P ) > α.

We assume by contradiction that

#u ∈ S1 : x(u, 0) ∈ Br(x) ∩ P ≤ p− 1 for all P ∈ A. (113)

Now we split x(·, 0) ∩Br(x) in two parts with big and small slope

U+ := u ∈ S1 : x(u, 0) ∈ Br(x), tan(∠(τ(u), L)) ≥ ∆4 + 1,

U− := u ∈ S1 : x(u, 0) ∈ Br(x), tan(∠(τ(u), L)) ≤ ∆4 + 1.

First we note that condition (113) implies the length estimate∫U−

dµ0 ≤ (1 + ∆4)(p− 1)H1(πLx(U−, 0)) ≤ 2r(1 + ∆4)(p− 1). (114)

To estimate the length of x(U+, 0) we consider the integral varifold Vt, de-fined by ∫

φ(x, S)dVt :=

∫S1

φ(x(u, t), Tx(u,t)x(S1, t))dµt,

for all φ ∈ C0c (Rn×G(n, 1),R). For a test function φ ∈ C2(Rn×G(n, 1),R)

we compute

d

dt

∫φ(x, S)dVt =

∫−φ|k|2 + dSφ(~ks + |k|2T )⊥ + dxφ(~k)dVt

=

∫−φ|k|2 − d2

Sφ(~k,~k)− dxdSφ(~k,~k) + dxφ(~k)dVt.

We choose a test function φ(x, S) = ϕ(x)ψ(S) where

ϕ(x) =

1 Br(x0),

0 BR(x0)c,

81

and |dϕ(x)| ≤ 2/(R− r), and ψ(S) = ψ(θ) with θ = ∠(L, S)

ψ(S) =

1 tan(θ) ≥ ∆4 + 1,

0 tan(θ) < ∆4,

and

|dψ(S)| ≤ 2/(tan−1(∆4 + 1)− tan−1(∆4))

≤ 2/(tan−1)′(∆4 + 1) = 2(1 + (1 + ∆4)2) ≤ 5,

and

|d2ψ(S)| ≤ 4/(tan−1(∆4 + 1)− tan−1(∆4))2 ≤ 4(1 + (1 + ∆4)2)2 ≤ 20.

With this choice we get the estimate∣∣∣∣ ddt∫φ(x, S)dVt

∣∣∣∣ ≤ (‖φ‖C0 + ‖dϕ‖C0‖dψ‖C0 + ‖d2ψ‖C0

) ∫DR(x0,t)

|k|2dµt

+ ‖dϕ‖C0

∫DR(x0,t)

|k|dµt

≤(

21 +10

R− r

)∫DR(x0,t)

|k|2dµt

+2

R− rµt(DR(x0, t))

1/2

(∫DR(x0,t)

|k|2dµt

)1/2

.

Since we will choose r < R < 1/4 and ε < 1, this implies∣∣∣∣ ddt∫φ(x, S)dVt

∣∣∣∣ ≤ C√Dε

R− r.

At time t = −1, since there are only Lipschitz graphs over L with slopesmaller than ∆4, we have∫

Dr(x0,0)

φ(x, S)dV−1 = 0.

Therefore, at time t = 0 we have∫U+

dµ0 ≤∫Dr(x0,0)

φ(x, S)dV0 ≤C√Dε

R− r.

Together with estimate (114) and since Dr(x0, 0) = U− ∪ U+ this implies

µ0(Dr(x0, 0)) ≤ 2r(1 + ∆4)(p− 1) +C√Dε

R− r.

This leads to the inequality

ε2 ≥∫ 0

−1

∫Dr(x0,t)

|k|2dµtdt

≥ µ−1(Dr(x0,−1))− µ0(Dr(x0, 0))

≥ 2rp− 2R(1 + ∆4)(p− 1)− C√Dε

R− r.

82

Once again we absorb the ε2 to arrive at

2rp− 2R(1 + ∆4)(p− 1) ≤ C√Dε

R− r. (115)

In the case p = 1 we choose R = 1/4 to obtain 1/2 ≤ 8C√Dε that leads to

a contradiction for ε0 small enough.

In the case p ≥ 2, we choose ε0 such that ∆4 ≤ 1/2(p − 1). Since p ≤ CDthis holds for ε0 = c/D5, where c > 0 is a constant depending on nothing.This actually determines the dependency of ε0 on D. With this choice thereholds p− (p− 1)(1 + ∆4) ≥ 1/2, hence we can choose

R =

(p

(p− 1)(1 + ∆4)+ 1

)r

2> r.

With this choice, the left-hand side of (115) is positive and the inequalitysimplifies to

r(p− (p− 1)(1 + ∆4)) ≤ C√Dε

R− r. (116)

Moreover there holds

r/4

(p− 1)(1 + ∆4)≤ R− r =

p− (p− 1)(1 + ∆4)

(p− 1)(1 + ∆4)

r

2≤ r

2.

The upper bound implies R ≤ 3r/2 = 3/16 which shows BR(x0) ⊂ B1 andactually justifies the computation above. By the lower bound of R − r wecan simplify inequality (116) further to

r2/8 ≤ C√Dε(p− 1)(1 + ∆4) ≤ C ′D3/2ε.

Recall that r = 1/8, hence this is a contradiction if we choose ε0 is smallenough.

J) Claim. The curvature is uniformly bounded by

supx(u,t)∈P1/6

|k(u, t)| ≤ Cε.

Proof. By item (I) in the ball B1/2 we have p independent components forall times t ∈ [−1, 0] and it is enough to get the estimate for one of them.By a rotation we can assume L = span(1, 0, ..., 0). Hence at time t = −1each component in B1/2 is the graph of a function w : [−1/2, 1/2] → Rn−1

with Lip(w) ≤ ∆4. Now we argue as in the proof of lemma 3.39. Inthis case the argument is simpler, since the flow in the ball Br1 is alreadysqueezed for all time in [−1, 0]. By theorem 3.11, for every (n − 1)−planeP with tan(∠(L,P )) > 3∆4 and P ∩ B1/2 ∩ N∆(L) 6= ∅ and all timest ∈ [−1, 0] there holds #x(·, t)−1(P ∩ Br1) ≤ p. This shows that eachcomponent in the ball B1/2 for every time t ∈ [−1, 0] is the graph of afunction r(·, t) : [−1/2, 1/2]→ Rn−1, with Lip(r(·, t)) ≤ 3∆4.

The function r(z, t) satisfies the graph flow equation (39) in the domain[−1/2, 1/2]× [−1, 0]. Hence, for ε0 small enough such that 3∆4 ≤ 1/10, by

83

theorem 3.30 and its proof, there exists a constant C depending on nothingsuch that

sup(z,t)∈P1/3

|k(z, t)|2 ≤ sup(z,t)∈P1/3

|rt(z, t)|2 ≤ C.

This curvature bound will allow us to use standard estimates for the linearparabolic equation. By the evolution equation (12) we have

(|k|2)t ≤ (|k|2)ss + 4|k|4,

and with ∂/∂s = (1 + |rz|2)−1/2∂/∂z we compute

(|k|2)t ≤(|k|2)zz1 + |rz|2

− 〈rz, rzz〉(1 + |rz|2)2

(|k|2)z + 4|k|4

=(|k|2)zz1 + |rz|2

− 〈rz, rt〉1 + |rz|2

(|k|2)z + 4|k|4

=: a(|k|2)zz + b(|k|2)z + c|k|2.

Note that there hold a ∈ [λ,Λ] for λ = 100/101, Λ = 1, |b| ≤ |rt| ≤ Cand |c| ≤ 4|k|2 ≤ 4C in the parabolic cylinder P1/3. By Lieberman [43]theorem 6.17 there exists a constant C depending only on λ, Λ, sup |b|/λand sup |c|/λ, hence in our case again on nothing, such that

sup(z,t)∈PR/2

|k(z, t)|2 ≤ C

R3

∫PR

|k(z, t)|2 dzdt

holds for R = 1/3. By multiplication of√

1 + |rz|2 ≥ 1 inside the integralthis implies

sup(z,t)∈P1/6

|k(z, t)|2 ≤ C∫ 0

−1

∫D1(t)

|k|2 dµtdt = Cε2.

Since x(u, t) = (z, r(z, t)) ∈ B1/6 implies z ∈ B1/6 this shows the requiredestimate

supx(u,t)∈P1/6

|k(u, t)|2 ≤ Cε2,

and hence proves the claim and the theorem.

We additionally rescale backwards to compute the dependency of the con-stants ε0 and C0 on the maximal area ratio D0. By item (C) and theargument at the end of item (D) we have

supPλ/6

|k| ≤ Cλ−2

∫Pλ

|k|2 dµtdt ≤ Cλ−2

∫P1

|k|2 dµtdt,

where λ = 2−CD/ε20 , ε0 = c/D5 and D = CD0. This leads to a constant of

the formC0(D) = Cλ−2 = C2CD

110 .

Again by the argument at the end of item (D) we obtain

ε0(D) = cε02−CD/ε20 = cD−5

0 2−CD110 ,

where 0 < c < C <∞ are constants depending on nothing.

84

4 Strip Pseudo-Distance

In this chapter we study the disk-size of a closed curve, that is the minimal areaof a spanning disk, and the strip pseudo-distance between two closed curves,which is the infimum of the area of annuli bounded by those curves.

The idea that these quantities are of interest for the curve shortening flow isfrom Altschuler and Grayson. In [8], they show how one can use Gauss-Bonnettheorem to prove that the disk-size of a evolving curve is a monotonically non-increasing quantity. By the same tick the strip pseudo-distance between twoevolving curves is non-increasing. In [49], Perelman used this idea for the curveshortening flow in a compact manifold (M, g(t)) that evolves according to theRicci flow. In this case the disk-size and strip pseudo-distance satisfies a growthrate estimate that depends on the curvature of the ambient manifold, see alsoMorgan and Tian [46]. Note that they stated their result only for ramp solu-tions, since their interest goes in a completely other direction. We will prove themonotonicity of the strip pseudo-distance in the second section. The argumentrequires an annulus type minimal surface on which Gauss-Bonnet theorem canbe applied. The existence of a minimal surface follows by Douglas [16] for dis-joint Jordan curves. In the general case we argue by an approximation. Such anapproximation exists, since smooth curves in high enough co-dimension generi-cally are disjoint and have no self-intersections. This is a geometrically obviousfact, and follows by Sards theorem, but the construction of the approximationis technically painful and we put it in the appendix C.

In the first section, we study the disk-size D0(x) and strip pseudo-distanceD0(x, y) for closed Lipschitz curves x, y, for itself without a flow. We alsodefine a disk-size D(x) and a strip pseudo-distance D(x, y) for Lipschitz arcsx, y. This allows us to split closed curves x, y with small D0(x, y) distance intosegments that are close in the D strip pseudo-distance. The definition of D willlooks somehow similar to the flat distance of the associated integral current,but in fact it is stronger than the flat distance. On the other hand are bothstrip pseudo-distances weaker than the C0 distance. Both D0 and D induces ametric on Lipschitz curves modulo parametrization and hairs, where a hair is aconnected segment of the curve that itself is a closed curve with zero disk-size.We will prove that in each equivalence class there exists a hairless representative,that is unique up to parametrization.

In the third section we will prove the following lower bound of the area ofdisk type surface

D0(x) ≥ c

M2,

spanned by a C2 curves with bounded curvature |k| ≤M , where c is a constantdepending on nothing. Even though this estimate seems elementary it is newto us. Moreover, we show the related estimate D0(x) ≥ cε2 for locally Lipschitzgraph-like curves with Lipε(x) ≤ 1. These estimates imply the intuitive factthat C2 curves as well as locally Lipschitz graph-like curves do no have hairs.

In the last section of this chapter we will use the lower area bound from aboveto prove that two locally Lipschitz graph-like curves that are close in the strippseudo-distance, in fact are Lipschitz close as maps modulo parametrization.This improves a lemma by Altschuler and Grayson [8] for ramps in R3.

85

4.1 Definition and Properties

Our aim is to define a disk-size and strip pseudo-distance separately for closedLipschitz curves and Lipschitz arcs. By a Lipschitz curve we mean a Lipschitzmap defined on an interval or S1. We define the length of a Lipschitz mapx : I → Rn by

L(x) :=

∫I

|xu(u)|du,

and denote the area of a Lipschitz map F : Ω ⊂ R2 → Rn by

A(F ) :=

∫Ω

√(DF )T (DF )dxdy.

Definition 4.1. For x ∈ C0,1(S1,Rn), we define the disk-size of a closed curveby

D0(x) := infF∈A

A(F ),

where A is the set of all Lipschitz maps F : B → Rn such that F |∂B = x, whereB denotes the 2−dimensional unit ball.

Definition 4.2. Let x, y ∈ C0,1(S1,Rn) be closed Lipschitz curves. We definethe strip pseudo-distance for closed curves by

D0(x, y) := infF∈A

A(F ),

where A is the set of all Lipschitz maps F : S1 × [0, 1] → Rn such that theboundary map satisfies F (u, 0) = x(u) and F (u, 1) = y(u).

Definition 4.3. For x ∈ C0,1([0, 1],Rn), we define the disk-size of a arc by

D(x) := infF∈A

√A(F ) +

L(∂F )− L(x)√4π

,

where A is the set of all Lipschitz maps F : B → Rn such that the boundarymap satisfies F (exp(πiu)) = x(u).

Definition 4.4. Let x, y ∈ C0,1([0, 1],Rn) be closed Lipschitz curves. We definethe strip pseudo-distance for arcs by

D(x, y) := infF∈A

√A(F ) +

L(∂F )− L(x)− L(y)√4π

,

where A is the set of all Lipschitz maps F : [0, 1] × [0, 1] → Rn such thatF (u, 0) = x(u) and F (u, 1) = y(u).

For closed curves we additionally define

Definition 4.5. For x, y ∈ C0,1(S1,Rn), we define

D0(x) :=√D0(x) and D0(x, y) :=

√D0(x, y).

86

Note that all definitions D0, D and D0 are invariant under reparametrization,and we may use them also for curves which are defined on closed interval.Moreover, by continuous continuation, we may use the definition of D also forcurves which are defined on open interval.

We will later proof that both D(x, y) and D0(x, y) define a pseudo-metric, i.e.satisfies all conditions of a metric without the property that D(x, y) = 0 impliesx = y. Since D0(x, y) is a pseudo-metric, also the squire-root D0(x, y) defines apseudo-metric. Note that D(x, y)2 does not satisfy the triangle inequality andhence is not a pseudo metric.

The disk-size and strip pseudo-distance D and D0 are related by the nextlemma. This chapter would slightly simplify if we only would study D and D0

without D0, but our main goal is to analyze the D0 pseudo-distance that fitstogether with that curve shortening flow.

Lemma 4.6. Let x, y ∈ C0,1([0, 1],Rn) be closed Lipschitz curves, i.e. x(0) =x(1) resp. y(0) = y(1), then there holds

D0(x) = D(x),

D0(x, y) ≤ D(x, y).

Proof. The inequality D0(x) ≥ D(x) follows directly from the definition. Forthe opposite inequality, let F be an admissible map for D(x). By definition ofD(x) there holds F (exp(πiu)) = x(u), and by assumption this is a closed curve.Hence also v(u) := F (exp(πi(u+ 1))), u ∈ [0, 1] is a closed curve.

We use the following isoperimetric inequality for Lipschitz maps from Alm-gren [2],

Theorem 4.7. (Optimal isoperimetric inequality for mappings) Suppose N isa compact (m+ 1)-dimensional Riemannian manifold with boundary. Then anyLipschitz map x : ∂N → Rn can be extended to a Lipschitz map F : N → Rnsuch that

Area(F ) ≤ γArea(x)(m+1)/m,

where γ =((m+ 1)|∂Bm+1

1 |1/m)−1

.

By theorem 4.7, we can extend F in the loop v such that

D0(v) ≤ L(v)2

4π.

This extension of F is an admissible map for D0(x) and we have the estimate

D0(x) ≤√A(F ) +D0(v)

≤√A(F ) +

√D0(v)

≤√A(F ) +

L(v)√4π

=√A(F ) +

L(∂F )− L(x)√4π

.

The inequality D0(x, y) ≤ D(x, y) follows similarly, since for an admissible mapF for D(x, y) the boundary map t 7→ F (0, t) together with t 7→ F (1,−t) definea closed curve.

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As we have seen in the proof above, additional closed loops do not affect thedisk-size. This allows us to extend the disk-size to piecewise Lipschitz curves,

S = x : [0, 1]→ Rn : x is piecewise Lipschitz .

Definition 4.8. Let x, y : [0, 1] → Rn. We denote by (−x) the curve with theopposite orientation,

(−x)(u) := x(1− u),

and by x ? y the concatenation of the curves

(x ? y)(u) :=

x(2u) if 0 ≤ u ≤ 1/2y(2u− 1) if 1/2 < u ≤ 1.

The concatenation ? defines an semi group structure on S which is notcommutative and has no identity element in S. Nevertheless, we will writex1 ? x2 ? x3 without brackets. This should not bother us, since ((x1 ? x2) ? x3)and (x1 ? (x2 ? x3)) are equal up to a piecewise linear homeomorphism, whichdoes not affect the disk-size and strip pseudo-distance. We extend the disk-sizeto x ∈ S that has from x = x1 ? ... ? xN , where xi : [0, 1] → Rn are Lipschitzcontinuous.

Definition 4.9. For x = x1 ? ... ? xN , where xi ∈ C0,1([0, 1],Rn), i = 1, ..., N ,the disk-size is define by

D(x) := infF∈A

√A(F ) +

L(∂F )− L(x)√4π

,

where A consists of all Lipschitz maps F : B → Rn, such that there existsvi ∈ C0,1([0, 1],Rn) such that the boundary map satisfies

F φ = x1 ? v1 ? ... ? xN ? vN ,

where φ(u) := exp(2πiu).

Note that the decomposition x = x1 ? ... ? xN is unique if we assume thatxi(1) 6= xi+1(0) for all i = 1, .., N − 1. On the other hand, if there holdsxi(1) = xi+1(0) the Lipschitz curve vi is a closed loop. By the same argumentas in the prove of lemma 4.6, vi does not affect D(x). This shows that D(x) iswell define, i.e. does not depend on the choice of the xi.

As an immediate consequence, we can write the D strip pseudo-distance viathe D disk-size. By a re-parametrization of F from [0, 1] × [0, 1] to ball B weget the following lemma.

Lemma 4.10. Let x, y ∈ C0,1([0, 1],Rn) be Lipschitz curves. Then there holds

D(x, y) = D(x ? (−y)).

Not every piecewise Lipschitz map x ∈ S is of the form x = x1 ? ... ? xN ,where xi : [0, 1]→ Rn are Lipschitz continuous. In particular, maps of this formare left continuous. It can be proven, that for every x ∈ S there exists a non-decreasing, surjective, piecewise linear map ψ : C0,1([0, 1], [0, 1]), and Lipschitzcurves xi ∈ C0,1([0, 1],Rn), i = 1, ..., N such that there holds

x := (x ψ)− = x1 ? ... ? xN ,

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where y− denotes the left limit of y : [0, 1] → Rn, i.e. y−(u) := limvu y(v),u > 0 and y−(0) := x(0).

For completeness we also define the disk-size for general x ∈ S.

Definition 4.11. For x ∈ S, we define the disk-size by

D(x) := infF∈A

√A(F ) +

L(∂F )− L(x)√4π

,

where A consists of all Lipschitz maps F : B → Rn, such that there existsa piecewise continuous, non-decreasing map θ : [0, 1] → [0, 1] such that theboundary map satisfies x = F φ θ, where φ(u) := exp(2πiu).

Remark. For our work it is not necessary to introduce the class S, but it maygive a better understanding. The attempt to make S into a vector space leadsto currents. We compare the strip pseudo-distance to the flat norm. To thisend, let T = [x] ∈ I1(Rn) be the associated integral 1-current to x ∈ S i.e.

T (w) :=

∫ 1

0

〈xu, w(x)〉 du = 0,

for each 1-form w ∈ D1(Rn).

Definition 4.12. Given an integral 1-current T ∈ I1(Rn), the flat norm is definedby

Flat(T ) = infF∈AM(F ) +M(S) ,

where A is the set of all integral 2-currents F ∈ I2(Rn) with ∂F = S + T , andM denotes the mass of a current.

Currents admit a vector space structure with scalar multiplication (λT )(w) :=λT (w) and addition (T + S)(w) := T (w) + S(w). The flat norm adds lengthand area together. This is necessary for the positive homogeneity Flat(λT ) =|λ|Flat(T ) of the flat norm. In the case of Lipschitz curves, there is no vec-tor space structure and the exponent of the area is chosen such that D(λx) =|λ|D(x) holds for space scaling.

Another difference is apparent in the degenerate case where x and y areconstant maps. In this case the flat distance Flat([x], [y]) vanishes, whereas√πD(x, y) is equal to the Euclidean distance between the points x, y. Since

in this case D0(x, y) also vanishes, this is also an example where the two strippseudo-distances D(x, y) resp. D0(x, y) do not agree.

Let x = x1 ? ... ? xI and T = [x] be the associated integral current. Theflat norm takes into account the orientation, but is independent of the orderin which the xi are added, since the associated current does not contain thisinformation. This is the main reason why the flat distance is unsuitable in thecontext of the curve shortening flow. Note that this lack of information fromthe associated current T = [x] occurs, if either x is not continuous or is notinjective. Hence the problem already exists for x ∈ C0,1([0, 1],Rn). See alsofigure 1 in the introduction. However, the dependency of the disk-size on theorder of the xi can be seen best for x ∈ S.

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Figure 3: Flat Distance vs Strip Pseudo-Distance

The example in figure 3 shows two closed curves where Flat([x], [y]) is smallwhereas D0(x, y) is not small. In the limit, when the curve segments piecewisecoincide, the flat distance vanishes. Since the curve shortening flow of the curvesdoes not coincide, the flat distance is not strong enough to imply uniqueness ofthe flow.

Sometimes we will use the following alternative definitions of the strip pseudo-distances. By a re-parametrization of F in definitions 4.1, 4.2 and 4.3 and weget the following three lemmata.

Lemma 4.13. Let x, y ∈ C0,1([0, 1],Rn). Then there holds

D0(x) = infF∈A

A(F ),

where A consists of all Lipschitz maps F : [0, 1]×[0, 1]→ Rn, such that F (u, 0) =x(u) and F (·, 1) = F (0, ·) = F (1, ·) = const.

Lemma 4.14. Let x, y ∈ C0,1([0, 1],Rn) be closed Lipschitz curves i.e. x(0) =x(1) and y(0) = y(1). Then there holds

D0(x, y) := infF∈A

A(F ),

where A consists of all Lipschitz maps F : [0, 1]×[0, 1]→ Rn, such that F (u, 0) =x(u), F (u, 1) = y(u) and F (0, t) = F (1, t).

Lemma 4.15. Let x ∈ C0,1([0, 1],Rn). Then there holds

D(x) = infF∈A

√A(F ) +

L(∂F )− L(x)√4π

,

where A consists of all Lipschitz maps F : [0, 1]× [0, 1]→ Rn such that x(u) =F (u, 0).

As mention above, (S, ?) is not a vector space and hence disk-size D(x) isnot a norm. However, the disk-size satisfies a triangle like inequality.

Lemma 4.16. There hold the triangle like inequalities

D(x) ≤ D(x, y) +D(y), x, y ∈ C0,1([0, 1],Rn),

D0(x) ≤ D0(x, y) +D0(y), x, y ∈ C0,1(S1,Rn).

We call these inequalities triangle like, since the first one, in view of lemma4.10, can be written as D(x) ≤ D(x ? (−y)) +D(y). In can be proven that thisinequality actually holds x, y ∈ S. Note that the inequality D(x, y) ≤ D(x) +D(y) only holds if x(1) = y(0) or y(1) = x(0), whereas D0(x, y) ≤ D0(x)+D0(y)holds generically.

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Proof. Let F1 : [0, 1] × [0, 1] → Rn be an admissible map of D(x, y) as indefinition 4.4, and let F2 : [0, 1]× [0, 1]→ Rn be an admissible map of D(y) asin lemma 4.15. Hence there holds F1(u, 1) = y(u) = F2(u, 0). Therefore, themap F (u, t) = (F1(u, ·) ? F2(u, ·))(t) continuous. Note that F is an admissiblemap for D(x) and hence there such that we have the estimate

D(x) ≤√A(F ) +

L(∂F )− L(x)√4π

=√A(F1) +A(F2) +

L(∂F1) + L(∂F2)− 2L(y)− L(x)√4π

≤√A(F1) +

√A(F2) +

L(∂F1)− L(y)√4π

+L(∂F2)− L(x)− L(y)√

4π.

Since this holds for all admissible maps F1 and F2, this proves the first inequalityof the lemma. The second inequality follows analogously, by the concatenationof admissible maps from the lemmata 4.14 and 4.13.

Proposition 4.17. The strip pseudo-distances D on C0,1([0, 1],Rn) and D0 onC0,1(S1,Rn) are pseudo-metrics, i.e. they satisfy D(x, x) = 0 resp. D0(x, x) =0, are symmetric and satisfy the triangle inequalities D(x, z) ≤ D(x, y)+D(y, z)resp. D0(x, z) ≤ D0(x, y) +D0(y, z).

Proof. The symmetry is obvious for both distances. D(x, x) = 0 and D0(x, x) =0 hold, since F (u, t) = x(u) is an admissible map for both.

To show the triangle inequality for the D-pseudo-distance, let F1 and F2

be admissible maps for the distances D(x, y) resp. D(y, z). Then F (u, t) =(F1(u, ·)?F2(u, ·))(t) is an admissible map for D(x, z) and we have the estimate

D(x, z) ≤√A(F ) +

L(∂F )− L(x)− L(z)√4π

≤√A(F1) +

L(∂F1)− L(y)− L(x)√4π

+√A(F2) +

L(∂F2)− L(y)− L(z)√4π

.

Since this holds for all admissible maps F1 and F2, this proves the triangleinequality for the D pseudo-distance.

The triangle inequality for the D0-pseudo-distance follows analogously.

As we have seen in lemma 4.10 the D strip pseudo-distance can be expressedby the disk-size via D(x, y) = D(x? (−y)). On the other hand, the disk-size canbe obtained as the degenerate case of the strip pseudo-distances. For a curvex ∈ C0,1([0, 1],Rn) there holds D(x) = D(x, y), where y ≡ x(0) or y ≡ x(1).

Similarly, for closed curves we have D0(x) = D0(x, y), where y ≡ p for somevector p ∈ Rn. In view of this fact the triangle inequality implies the trianglelike inequality

D0(x) ≤ D0(x, y) +D0(y),

as well asD0(x, y) ≤ D0(x) +D0(y).

Another consequence of the triangle inequality is the following lemma.

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Lemma 4.18. Let x, y, z ∈ C0,1([0, 1],Rn). If x(1) = y(0) and y(1) = z(0),then there holds

D(x ? y,−z) = D(x, (−z) ? (−y)), (117)

andD(x ? y, y) = D(x). (118)

Let x1, x2, y1, y2 ∈ C0,1([0, 1],Rn), and assume x1(1) = x2(0) and y1(1) = y2(0),then there holds

D(x1 ? x2, y1 ? y2) ≤ D(x1, y1) +D(x2, y2). (119)

Proof. The first equality follows directly from lemma 4.10 via

D(x ? y,−z) = D(x ? y ? z) = D(x,−(z ? y)) = D(x, (−z) ? (−y)).

Now we prove the third inequality. Let F1, F2 : [0, 1]×[0, 1]→ Rn be admissi-ble maps for D(x1, y1) resp. D(x2, y2). Let F (u, t) := (F1(·, t)?F2(·, t))(u). Themap F may have a hole with boundary w = F1(1, ·)?(−F2(0, ·)). By the isoperi-metric inequality, we can extend F to an admissible map for D(x1 ? x2, y1 ? y2)such that there holds

D(x1 ? x2, y1 ? y2) ≤√A(F ) +D0(w) +

L(∂F )− L(x1 ? x2)− L(y1 ? y2)√4π

≤√A(F ) +

L(w)√4π

+L(∂F )− L(x1 ? x2)− L(y1 ? y2)√

4π

≤√A(F1) +

L(∂F1)− L(x1)− L(x2)√4π

+√A(F2) +

L(∂F2)− L(y1)− L(y2)√4π

.

Inequality (119) follows, since this holds for all admissible maps F1 and F2.The inequality D(x) ≤ D(x ? y, y) follows by the triangle like inequality

D(x) ≤ D(x, y ? (−y)) +D(y ? (−y)) = D(x ? y, y) +D(y, y) = D(x ? y, y).

The opposite inequality follows by (119) via

D(x ? y, y) = D(x ? y, x(1) ? y) ≤ D(x, x(1)) +D(y, y) = D(x).

Now we compare the strip pseudo-distance of Lipschitz curves with the C0

norm of their parameterizations.

Lemma 4.19. Let U be either [0, 1] or S1 and let x, y ∈ C0,1(U,Rn) be Lipschitzcurves. If ‖x− y‖C0 < 1, then the strip pseudo-distances are bounded by

D(x, y)2 ≤ C (L(x) + L(y) + 1) ‖x− y‖C0 ,

resp.D0(x, y) ≤ C (L(x) + L(y) + 1) ‖x− y‖C0 .

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Proof. We can assume that ε = ‖x−y‖C0 > 0. Let v(u) := (y(u)−x(u))/ε ∈ B1

and define the surfaceF (u, t) := x(u) + tv(u),

for (u, t) ∈ S1 × [0, ε]. With dF = (xu + tvu, v) we compute the area

A(F ) =

∫ ε

0

∫U

√(xu + tvu)2 + v2 − ((xu + tvu)v)2dudt

≤∫ ε

0

∫U

(2|xu + tvu|+ 1) dudt

≤ 2εL(x) + ε2

∫U

|vu|du+ 2πε

≤ 2εL(x) + ε(L(x) + L(y)) + 2πε.

The last inequality follows from

ε

∫I

|vu|du ≤∫U

|xu|du+

∫U

|xu + εvu|du = L(x) + L(y).

This shows the estimate for the D0-pseudo-distance. For the D-pseudo-distancewe additionally use the estimates L(F (0, ·)), L(F (1, ·)) ≤ ε. Hence we get thebound

D(x, y)2 ≤ C(A(F ) + (L(F (0, ·)) + L(F (1, ·))2

)≤ C(ε+ ε2).

Therefore, as long as the lengths of the curves are bounded, uniform conver-gence implies that the strip pseudo-distances converge.

Corollary 4.20. Let U be either [0, 1] or S1. Let x, y, z, w ∈ C0,1(U,Rn) beLipschitz curves. For small C0 distances we have the estimates

|D(x)−D(y)| ≤ C‖x− y‖1/2C0 ,

|D(x, y)−D(z, w)| ≤ C(‖x− z‖1/2C0 + ‖y − w‖1/2C0

),

resp. for closed curves there holds

|D0(x)−D0(y)| ≤ C‖x− y‖C0 ,

|D0(x, y)−D0(z, w)| ≤ C (‖x− z‖C0 + ‖y − w‖C0) ,

where C only depends on the lengths L(x), L(y), L(z) and L(w).

Proof. By the triangle like inequality of lemma 4.16 we have

|D(x)−D(y)| ≤ D(x, y),

and the triangle inequality implies

|D(x, y)−D(z, w)| ≤ D(x, z) +D(y, w).

The same inequalities hold for closed curves and the D0-pseudo-distance. Hencethe corollary follows immediately from the lemma above.

93

Remark. We give an example to show that the length bound is necessary. Infact, for all ε > 0 and C > 0 there exist curves x, y ∈ C∞(S1,Rn) such that‖x− y‖∞ ≤ ε and D(x, y) ≥ C.

This happens if the curves are winding around each other. For example,consider the helix x(u) = (ε cos(u/δ), ε sin(u/δ), u) and y(u) = (0, 0, u) for u ∈[0, 2π]. Then x, y are ε-close in the C0-norm. If we connect x, y by two additionallines we get a closed curve. The disk type minimal surface spanned by this closedcurve is a part of a helicoid and the area is given by

A = π(εδ

√ε2 + δ2 + δ sinh−1

(εδ

)).

If we choose δ = πε2/C then the area is bounded from below by A ≥ C. Withthis choice the length of x becomes unbounded L(x) = 2π

√ε2 + δ2/δ ≥ 2C/ε.

In this example x, y are not closed curves, but we can choose a big circle for yand a helix in a torus for x.

Now we are in the position to prove, that the flat norm is bounded by thestrip pseudo-distance.

Lemma 4.21. Let x ∈ C0,1(S1,Rn) be a Lipschitz curve and let T = [x] ∈I1(Rn) be the associated integral current. Then there holds

Flat(T ) ≤ D0(x).

Proof. Let xi ∈ C∞(S1,Rn) be an approximation of x in C0 and W 1,p, forp < ∞. By convergence in W 1,1 the corresponding integral current convergesTi T i.e.

Ti(w) :=

∫S1

⟨xiu, w(x)

⟩du→ T (w).

Since xi are closed C∞ curves, we have ∂Ti = 0. Therefore, there exist min-imizing integral currents Si ∈ I2(Rn) such that ∂Si = Ti. By the minimizingproperty of Si we have the mass bound M(Si) ≤ D0(xi). By the triangle in-equality and lemma 4.20 there holds

Flat(Ti) = M(Si) ≤ D0(xi) ≤ D0(xi, x) +D0(x)→ D0(x), (i→∞).

The pseudo-metrics give rise to metric spaces by metric identification.

Definition 4.22. We define the equivalence relations on C0,1([0, 1],Rn)

x ∼ y ⇔ D(x, y) = 0,

and on C0,1(S1,Rn)x ∼0 y ⇔ D0(x, y) = 0.

The symmetry and triangle inequality of the strip pseudo-distances imply thesymmetry and transitivity of the relations, and by D(x, x) = 0 resp. D0(x, x) =0 the relations are also reflexive.

The goal of this section is to describe the metric spaces(C0,1([0, 1],Rn)/ ∼, D

),(

C0,1(S1,Rn)/ ∼0, D0

).

To analyze the equivalence relations we make the following definition.

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Definition 4.23. Let U be either [0, 1] or S1. Let x ∈ C0,1(U,Rn) be a Lipschitzcurve. A hair of x is an open connected interval I ⊂ U such that D(x|I) = 0.

Recall that the disk-size D(x|I) for a map x defined on an open interval Iis given by continuous continuation.

Remark. An example of a hairs is x?(−x), where x is a Lipschitz curve. Anotherexample is x ? y ? (−y) ? z ? w ? (−w) ? (−z) ? (−x), as long as this defines acontinuous curve. Hairs have always a point where the curve turns instantly inthe opposite direction.

By definition a hair is a part of a curve that the strip pseudo-distances donot see. Since D(x|I) = 0 implies that x|I is a closed curve, we may add orsubtract them from a closed curve. We will show that an equivalence class [x] of∼ resp. ∼0 consists precisely of the curves modulo hairs and parametrization.Moreover, we show that there exists a hairless representative, which is uniqueup to parametrization.

It seems that the equivalence relations ∼ resp. ∼0 are very similar. But fora closed curve x ∈ C0,1([0, 1],Rn) with x(0) = x(1), it may happen that thereappears an additional hair in S1.

In the definition of a hair I, it is important that the interval is connected.Otherwise, we have to consider x|I as an element of S. Clearly, D(x) = 0 stillimplies that x|I is a closed curve, but we lose control over the order in whichthe curve is traversed. This would correspond to parts of a curve, that the flatdistance does not see.

We have shown in lemma 4.21 that Flat([x]) ≤ D0(x) = D(x)2. The op-posite inequality does not hold in general. The smooth closed curve shown infigure 4 is a counterexample. In this example Flat([x]) correspond to a genusone surface. In the limit where the curves piecewise coincide, the flat normFlat([x]) vanishes whereas D0(x) corresponds to two disks which coincides.

Figure 4: Flat Norm vs Disk-Size

The third inequality of lemma 4.18 can also be written as

D(x, y) ≤ D(x|[0, a], y|[0, b]) +D(x|[a, 1], y|[b, 1]),

where 0 < a, b < 1. Under certain assumptions, the right hand side can also beestimated in terms of the left hand side.

Lemma 4.24. Let x, y ∈ C0,1([0, 1],Rn) be two Lipschitz curves such thatD(x, y) < D(x). Then there exist 0 < a, b < 1 such that L(x|[0, a]), L(y|[0, b]) ≥√π(D(x)−D(x, y))/2 and analogously for [a, 1] and [b, 1], and

D(x|[0, a], y|[0, b]) ≤ D(x, y) +D(x, y)2

2π(D(x)−D(x, y)),

95

and analogously for D(x|[a, 1], y|[b, 1]).

Proof. We first approximate by smooth curves. By mollification we find xi, yi ∈C∞([0, 1],Rn) which converge to x, y in C0 and W 1,1. Since the length of thecurves converges, corollary 4.20 implies that the strip pseudo-distances convergeand there holds D(xi, yi)

2 ≤ C (‖xi − x‖C0 + ‖yi − y‖C0), where the constantC only depends on L(x) and L(y).

Let ε > 0. Again by approximation we find Fi ∈ C∞([0, 1]× [0, 1],Rn), withFi(u, 0) = xi(u), Fi(u, 1) = yi(u) and zi(t) = Fi(0, t), wi(t) = Fi(1, t) such that

D(xi, yi) ≤√A(Fi) +

L(zi) + L(wi)√4π

≤ D(xi, yi) + ε.

We consider Q = [0, 1]× [0, 1] with the pullback metric (gi)kl = Dk(Fi)jDl(Fi)

j .Let ∆i = distgi(0 × [0, 1], 1 × [0, 1]) and let t1, t2 such that

∆i = distgi((0, t1), (1, t2)).

Together with zi|[0, t1] and wi|[0, t2] we find a path connecting (0, 0) to (1, 0).This shows the estimate D(xi) ≤ D(xi, yi) + ∆i/

√4π. In the limit this leads to

lim inf ∆i ≥√

4π(D(x)−D(x, y)) > 0.Therefore, we can assume that ∆i > 0, and use the co-area formula with the

function f(u, t) := distgi((u, t), 0 × [0, 1]) to get∫ ∆i

0

Lengthgi(f = l)dl ≤∫Q

|Df |dµi ≤∫Q

dµi = A(Fi),

where µi is the measure introduced by gi. There exists a value δi ∈ [∆i/4, 3∆i/4]such that Lengthgi(f = δi) ≤ 2A(Fi)/∆i. Since f is continuous f = δiconnects xi to yi. Hence there exist 0 ≤ ai, bi ≤ 1 such that

distgi((ai, 0), (bi, 1)) ≤ 2A(Fi)/∆i.

Therefore, we have the estimate

D(xi|[0, ai], yi|[0, bi]) ≤√A(Fi) +

L(zi)√4π

+2A(Fi)√

4π∆i

≤ (D(xi, yi) + ε) +(D(xi, yi) + ε)2

√π∆i

.

Now we converge sub-sequentially ai → a, bi → b and ε → 0. Since thecurves converge in C0, by corollary 4.20 there holds

D(x|[0, a], y|[0, b]) ≤ D(x, y) +D(x, y)2

2π(D(x)−D(x, y)).

Moreover, by construction we have

L(xi|[0, ai]) ≥ δi ≥ ∆i/4,

and by convergence in W 1,1 this implies L(x|[0, a]) ≥√π(D(x) − D(x, y))/2.

The other inequalities follows similarly and we conclude 0 < a, b < 1.

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We will use the lemma above only in the following way.

Corollary 4.25. Let x, y ∈ C0,1([0, 1],Rn) be two Lipschitz curves such thatD(x) > 0 and D(x, y) = 0. Then there exist 0 < a, b < 1 with the followingproperties,

1) x(a) = y(b),

2) L(x|[0, a]), L(y|[0, b]) ≥√πD(x)/2 and analogously for [a, 1] and [b, 1],

3) D(x|[0, a], y|[0, b]) = 0 and D(x|[a, 1], y|[b, 1]) = 0.

Remark. The assumption and conclusion of the corollary are symmetric in x, ysince D(x, y) = 0 implies D(x) = D(y).

Analogously to the lemma above, under suitable assumptions, there exists akind of reverse estimate to the inequality D0(x, y) ≤ D(x, y).

Lemma 4.26. Let x, y ∈ C0,1(S1,Rn) be two closed Lipschitz curves. IfD0(x, y) < D0(x), then there exist 0 ≤ a, b ≤ 1 such that the parameteriza-tions x(u) = x(exp(2πi(u + a)))) and y(u) = y(exp(2πi(u + b))) for u ∈ [0, 1]satisfy the inequality

D(x, y) ≤√D0(x, y) +

D0(x, y)

2π√D0(x)−D0(x, y)

.

Proof. Analogously to the last lemma, we approximate x, y with smooth curvesxi, yi ∈ C∞(S1,Rn) in C0 and W 1,1 and the strip pseudo-distance by a smoothmap Fi : S1 × [0, 1]→ Rn with Fi(u, 0) = xi(u), Fi(u, 1) = yi(u) such that

D0(xi, yi) ≤ A(Fi) ≤ D0(xi, yi) + ε.

We consider Q = S1 × [0, 1] with the pullback metric (gi)kl = Dk(Fi)jDl(Fi)

j .Let ∆i := distgi(S

1 × 0, S1 × 1). We cut the curves xi, yi at the pointspi, qi ∈ S1, where the distance ∆i is attained, xi(u) = xi(exp(2πi(u + ai)))and yi(u) = xi(exp(2πi(u + bi))), u ∈ [0, 1], and cut the annulus Q along thegeodesic between pi and qi. This yields the estimate


2∆i√4π.

If lim inf ∆i = 0, we take a sub-sequential limit ai → a, bi → b. Hence xi → x inC0 and analogously for y. Together with the limit ε → 0 this shows D(x, y) ≤√D0(x, y).If ∆ = lim inf ∆i > 0, we can use the coarea formula with the function

f(u, t) = distgi((u, t), S1 × 0), to conclude∫ ∆i

0

Lengthgi(f = l)dl ≤ A(Fi).

Hence, there exists a δi ∈ (0,∆i) such that

Lengthgi(f = δi) ≤ A(Fi)/∆i.

97

We use the isoperimetric inequality to extend Fi inside the closed curve f = δi.Together with the length bound this implies

D0(xi) ≤ A(Fi) +D0(f = δi)

≤ A(Fi) +Lengthgi(f = δi)2

4π

≤ A(Fi) +A(Fi)

2

4π∆2i

.

Since by assumption D0(x, y) < D0(x), we choose ε > 0 small enough such thatA(Fi) ≤ D0(xi, yi) + ε < D(xi). This leads to the estimate

∆2i ≤

A(Fi)2

2π(D0(xi)−A(Fi)).

Since ∆i = distgi(pi, qi) we conclude


2∆i√4π

≤√A(Fi) +

A(Fi)

2π√D0(xi)−A(Fi)

.

In the limit this implies the inequality

D(x, y) ≤√D0(x, y) +

D0(x, y)

2π√D0(x)−D0(x, y)

.

Lemma 4.27. Let U be either [0, 1] or S1. Let x, y ∈ C0,1(U,Rn) be two Lips-chitz curves which do not contain hairs. If they have zero strip pseudo-distanceD(x, y) = 0 resp. D0(x, y) = 0, then they are equal up to parametrization.

Proof. We first reduce the case U = S1 to the case U = [0, 1]. Since by assump-tion D0(x) = D(x)2 > 0 and D0(x, y) = 0, we can use lemma 4.26 to find mapsx, y ∈ C0,1([0, 1],Rn) such that D(x, y) = 0.

Now we consider the case U = [0, 1]. The goal is to find a bijection φ :[0, 1] → [0, 1], φ(a) = b such that x = y φ. Since D(x, y) = 0, the start andendpoints agree, and we can define φ(0) = 0 and φ(1) = 1. Let 0 < a < 1. Sincex does not contain hairs, D(x|I) > 0 for each open interval I. Therefore, wecan use corollary 4.25 to find 0 < a1, b1 < 1 such that x(a1) = y(b1). If a < a1

we defineI1 = [0, a1] and J1 = [0, b1].

If a1 < a we defineI1 = [a1, 1] and J1 = [b1, 1].

In case of a1 = a we define Ik = a and Jk = b for all k ≥ 1. Other-wise, since by corollary 4.25 D(x|I1, y|J1) = 0, we can iterate this procedureto find sequences ak, bk with x(ak) = y(bk) and closed intervals Ik+1 ⊂ Ik andcorresponding intervals Jk+1 ⊂ Jk such that D(x|Ik, y|Jk) = 0 and

L(x|Ik+1) ≤ L(x|Ik)−√πD(x|Ik)

2, (120)

L(y|Jk+1) ≤ L(y|Jk)−√πD(y|Jk)

2. (121)

98

By monotonicity of the sequences, I = ∩k≥1Ik and J = ∩k≥1Jk are non-empty, closed intervals. The sequences L(x|Ik) and L(y|Jk) are monotone de-creasing and bounded, hence converge. Therefore, inequalities (120) and (121)imply that D(x|I) = 0 and D(y|J) = 0. Since we assume that there are no hairs,the intervals consist only of one point I = a resp. J = b. By constructionwe have ak ∈ Ik and bk ∈ Jk, hence lim ak = a and lim bk = b. We defineφ(a) := b and conclude x(a) = y(φ(a)).

Claim. There holds D(x|[0, a], y|[0, b]) = 0 = D(x|[a, 1], y|[b, 1]).We prove by induction that D(x|[0, ai], y|[0, bi]) = 0 for all i ≥ 1. For i = 1 thisfollows from corollary 4.25. Assume that the statement holds for all 1 ≤ i ≤ k.

In case ak < a < ak+1, it follows that bk < b < bk+1 and Ik+1 = [ak, ak+1],Ik = [ak, ai] for some 1 < i < k. By lemma 4.18, we have the estimate

D(x|[0, ak+1], y|[0, bk+1]) ≤ D(x|[0, ak], y|[0, bk]) +D(x|Ik+1, y|Jk+1).

The first term on the right hand side vanishes by induction and the second sinceD(x|Ik, y|Jk) = 0 holds for all k ≥ 1.

In case ak+1 < a < ak, it follows that Ik+1 = [ak+1, ak], Ik = [ai, ak] forsome 1 < i < k and we have the estimate

D(x|[0, ak+1], y|[0, bk+1]) ≤ D(x|[0, ai], y|[0, bi]) +D(x|[ai, ak+1], y|[bi, bk+1])

= D(x|[0, ai], y|[0, bi]) +D(x|Ik \ Ik+1, y|Jk \ Jk+1).

The first term on the right hand side vanishes again by induction and the secondsince, by construction and corollary 4.25, we have D(x|Ik \Ik+1, y|Jk \Jk+1) = 0for all k ≥ 1.

In case ak < ak+1 < a, we have Ik+1 = [ak+1, ai], Ik = [ak, ai] for some1 < i < k and thus

D(x|[0, ak+1], y|[0, bk+1]) ≤ D(x|[0, ak], y|[0, bk]) +D(x|Ik \ Ik+1, y|Jk \ Jk+1).

If a < ak+1 < ak, it follows that Ik+1 = [ai, ak+1], Ik = [ai, ak] for some1 < i < k and there holds

D(x|[0, ak+1], y|[0, bk+1]) ≤ D(x|[0, ai], y|[0, bi]) +D(x|Ik+1, y|Jk+1).

This completes the induction. By convergence in C0 and corollary 4.20, thisimplies the first inequality of the claim. The second inequality follows by sym-metry.

Claim. The map φ is well defined, i.e. the limit b is independent of thesequences ak, bk.Proof. Let ck, dk be different sequences converging to a, d. We may assumethat b ≤ d. We use lemma 4.16 and since x(0) = y(0), x(1) = y(1) andy(b) = x(a) = y(d) we can again use lemma 4.18 to get the estimate

D(y|[b, d]) ≤ D(y|[b, d], (−y|[0, b]) ? x ? (−y|[d, 1]))

+D((−y|[0, b]) ? x ? (−y|[d, 1]))

= D(y|[0, b] ? y|[b, d] ? y|[d, 1], x) +D(x, y|[0, b] ? y|[d, 1])

≤ D(x, y) +D(x|[0, a], y|[0, b]) +D(x|[a, 1] ? (−y|d, 1])).

All terms on the right hand side vanish and since y does not contain hairs thisshows b = d.

99

Claim. The map φ is monotonically increasing.First, we mention that if 0 < a < 1 then the sequence ak is alternating aroundthe limit a, i.e. if ak ≤ a then there exists l > k such that a ≤ al andanalogously for the opposite inequalities. This follows by contradiction. If weassume that ak a then Ik = [0, ak] which converge to a hair I = [0, a]. Sinceby construction a < ak is equivalent to b < bk, the sequence bk is also alternatingaround the limit.

Let a < c, b = φ(a) and d = φ(c). Let ak, bk be sequences, as in theconstruction, converging to a, b. By the previous argument there exists anak with a ≤ ak < c. Therefore, by uniqueness and construction, we haveb ≤ bk < d.

Furthermore, the map is surjective. This follows by symmetry, since y doesnot contain hairs, we can fix 0 < b < 1 and find 0 < a < 1 with x(a) = y(b).Thus, the curves are equal up to parametrization y = x · φ.

Definition 4.28. Let U be either [0, 1] or S1 and x ∈ C0,1(U,Rn) a Lipschitzcurve.

1) H ⊂ U is a hair collection of x, if H = ∪Ii is a disjoint union of hairs Ii.

2) H is a maximal hair collection, if there is no hair collection H ′ such thatH ( H ′.

Note that a maximal hair collection is not unique. Since S1 and [0, 1] areseparable, a hair collection is a countable union of hairs.

Lemma 4.29. Let U be either [0, 1] or S1 and let H = ∪Ii ⊂ U be a maximalhair collection of x ∈ C0,1(U,Rn). Then the hairs satisfy dist(Ii, Ij) > 0 fori 6= j. Moreover, if D(x) 6= 0 then Hc contains an open subset.

Proof. If I = (u1, u2) and J = (v1, v2) are disjoint hairs with dist(I, J) = 0, wecan assume that u2 = v1, and it follows that D(x|(u1, v2)) ≤ D(x|I) +D(x|J) =0. This contradicts the maximality of H since we can replace I and J by theinterval (u1, v2).

By the argument above, if Hc does not contain an open subset, then Hconsists only of one hair. In case U = [0, 1] this hair is H = (0, 1) and if U = S1

the hair is H = S1 \ p. Hence it follows D(x) = 0.

Now we show that there exists a hairless representative.

Lemma 4.30. Let U be either [0, 1] or S1. Let x ∈ C0,1(U,Rn) be a Lipschitzcurve that is not a hair i.e. D(x) 6= 0. Then there exists a hairless Lipschitzcurve x ∈ C0,1(U,Rn) such that D(x, x) = 0 resp. D0(x, x) = 0. Moreover, theLipschitz curve x is unique up to parametrization.

Remark. The case of a hair D(x) = 0 we have already studied at the beginningof this chapter. Every curve x with D(x, x) = 0 resp. D0(x, x) = 0 also satisfiesD(x) = 0 and hence has the hair I = (0, 1) resp. I = S1 \ p. In this case, asuitable representative is the constant map x ≡ x(0) resp. x ≡ 0 ∈ Rn.

Proof. We consider the case U = [0, 1]. Let H = ∪i≥1Ii be a maximal haircollection with hairs Ii = (ai, bi).

100

We denote by Hk = ∪ki=1(ai, bi], Lk = H1(Hk), li = H1(Ii) = bi − ai > 0and L = H1(H). We can assume that L > 0 since otherwise we have H = ∅.By lemma 4.29 we also have L < 1.

Now we define maps to cancel the hairs successively. For convenience wedenote by φ0 = id[0,1] and recursively define the functions φk : [0, 1 − Lk] → Rby

φk(u) =

φk−1(u) if 0 ≤ u ≤ Akφk−1(u+ lk) if Ak < u ≤ 1− Lk,

where Ak = H1([0, ak] ∩Hck).

Claim. The maps φk are monotone. Moreover, if u > v then φk(u)−φk(v) ≥u− v.Proof by induction. For φ0 this is trivial and the inequality φk(u) − φk(v) ≥φk−1(u) − φk−1(v) follows from the definition of φk and the monotonicity ofφk−1.

Claim. The maps φk jump at the point Aik := H1([0, ai] ∩Hck) by the value

li = H1(Ii), 1 ≤ i ≤ k. Moreover, we have the explicit formula

φk(u) = u+∑Aik<u

li.

Proof by induction. Clearly, φk jumps at Ak = Akk by the value li. Now weconsider 1 ≤ i < k. For Aik < Ak we have ai < bi < ak and hence Aik = Aik−1.For Ak < Aik we have ak < bk < ai and this implies

Aik = H1([0, ai] ∩Hck−1 ∩ Ick) = Aik−1 − li.

Claim. The maps φk : [0, 1 − Lk] → [0, 1] \ Hk are bijections, with inversemaps

ψk(v) := H1([0, v] ∩Hck).

Proof. For v ∈ [0, 1] \Hk we have

φk(ψk(v)) = ψk(v) +∑

Aik<ψk(v)

li = H1([0, v] ∩Hck) +H1([0, v] ∩Hk) = v.

Since ψk(ak) = Ak this implies φk−1(Ak) = φk(Ak) = ak and φk−1(Ak + lk) =bk. Now we show ψk(φk(u)) = u by induction.

ψk(φk(u)) =

H1([0, φk−1(u)] ∩Hc

k) if 0 ≤ u ≤ AkH1([0, φk−1(u+ lk)] ∩Hc

k) if Ak < u ≤ 1− Lk.

By monotonicity u ≤ Ak implies φk−1(u) ≤ φk−1(Ak) = ak and Ak < u impliesφk−1(u+ lk) > φk−1(Ak + lk) = bk. Therefore, there holds

ψk(φk(u)) =

H1([0, φk−1(u)] ∩Hc

k−1) if 0 ≤ u ≤ AkH1([0, φk−1(u+ lk)] ∩Hc

k−1)− lk if Ak < u ≤ 1− Lk.

Since by assumption ψk−1(φk−1(u)) = u this shows the claim.Claim. φk converges pointwise to some map φ : [0, 1 − L] → [0, 1] \H, and

ψk converges uniformly to a Lipschitz map ψ : [0, 1] → [0, 1 − L]. Moreover,there holds ψ φ = id[0,1−L].

101

Proof. Since φk−1(u) ≤ φk−1(u + lk), by definition of φk we have φk(u) ≥φk−1(u). Hence, the limit φ(u) = limk φk(u) exists for each u ∈ [0, 1−L]. Sinceall φk(u) are contained in the closed set [0, 1]\H, also the limit φ(u) is containedin this set.

The maps ψk : [0, 1] → [0, 1 − Lk], ψk(v) := H1([0, v] ∩ Hck) are uniformly

Lipschitz continuous with Lip(ψk) ≤ 1. Since Hk ⊂ Hk+1 there holds ψk(u) ≥ψk+1(u). Hence ψk converges pointwise to a Lipschitz map ψ. Since Lip(ψ) ≤ 1we have

|ψk(φk(u))− ψ(φ(u))| ≤ |ψk(φk(u))− ψk(φ(u)|+ |ψk(φ(u))− ψ(φ(u))|≤ |φk(u)− φ(u)|+ |ψk(φ(u))− ψ(φ(u))|.

This proves the claim since ψk(φk(u)) = u and the right hand side converges tozero as k tends to infinity.

We rescale this sequences to a fixed domain φk : [0, 1]→ [0, 1] \Hk, by

φk(u) = φk((1− Lk)u),

which converges to φ(u) = φ((1− L)u) and ψk : [0, 1]→ [0, 1] by

ψk(u) = (1− Lk)−1ψk(u),

which converges to ψ(u) = (1 − L)−1ψ(u) with ψ φ = id[0,1]. Now we definethe sequences of maps xk : [0, 1− Lk]→ Rn, k ≥ 0 by

xk(u) = x(φk(u)),

and xk : [0, 1]→ Rn, byxk(u) = x(φk(u)).

Note that there holds x0 = x0 = x.Recall that φk(Ak) = φk−1(Ak) = ak and lim

uAkφk(u) = lim

uBkφk−1(u) = bk,

where Bk = Ak + lk. Since the start and endpoint of a hair coincide, we have

xk(Ak) = x(ak) = x(bk) = limuAk

x(φk(u)) = limuAk

xk(u).

Therefore, the maps xk, xk are continuous. Furthermore, we have

L(xk) = L(xk) ≤ L(x),

andLip(xk) = (1− Lk)−1Lip(xk) ≤ Lip(xk) ≤ Lip(x).

In particular, the sequences are uniformly bounded and equicontinuous, henceby Arzela-Ascoli there exist convergent subsequences, without relabeling wehave xk → x and xk → x in C0.

SinceD(xk−1|[0, Ak], xk|[0, Ak]) = 0, D(xk−1|[Bk, 1−Lk−1], xk|[Ak, 1−Lk]) =0 and D(xk−1|Ik) = 0, by lemma 4.18 there holds D(xk−1, xk) = 0. Sincethe strip pseudo-distance is invariant under re-parametrization, we also haveD(xk−1, xk) = 0. By the triangle inequality we conclude that

D(x, xk) = D(x0, xk) ≤k∑i=1

D(xi−1, xi) = 0,

102

for all k ≥ 0. By corollary 4.20 the strip pseudo-distance converges and we get

D(x, x) = limk→∞

D(x, xk) = 0.

Now we show that the curve x is hairless. By contradiction, let J = (c, d) bea hair of x. By definition of φ there holds a := φ(c), b := φ(d) /∈ H. We definethe open interval I = (a, b) ⊂ [0, 1]. Since the boundary point of I are not in H,the set H ∩ I is a maximal hair collection of the curve y = x|I. Let yk = xk|Jk,where Jk = ψk(I) = (ck, dk). The Jk are chosen such that boundary points ofyk are constant, yk(ck) = x(φk(ck)) = x(a) and analogously for dk.

Now we show that yk converges uniformly to y = x|J . Since Jk = ψk(I) =(ck, dk) → J = ψ(I), for every u ∈ J there exists a K such that for all k ≥ K,u ∈ Jk. For such k ≥ K we have the estimate

|yk(u)− y(u)| = |xk(u)− x(u)| ≤ ‖xk − x‖C0 .

Analogously to the construction above, the uniform convergence impliesD(y, y) = 0, and since by assumption D(y) = 0, we conclude D(y) ≤ D(y, y) +D(y) = 0. Hence, I is a hair of x which contradicts the maximality of H.

It remains to show that the restriction is unique up to re-parametrization.This follows directly from lemma 4.27 since x is hairless and D(x, x) = 0.

Now we reduce the case U = S1 to the case U = [0, 1]. Since D(x) 6= 0by lemma 4.29 there exists U ⊂ S1 \H. Therefore, we find a parametrizationx : [0, 1] → Rn with x(0) = x(1) and such that H ⊂ [0, 1] is separated fromthe boundary point. Hence, the construction above does not change x near theboundary point. This is necessary to make sure that there does not occur a hairof x as a closed curve.

Finally, by lemma 4.6 we have D0(x, x) ≤ D(x, x)2 = 0.

Remark. Maybe there exists another existence proof by minimization since therestriction to the hairless part x minimizes the length under the constraintD(x, x) = 0.

4.2 Monotonicity of the Strip Pseudo-Distance

First, we show that a curve shortening flow is uniformly Lipschitz continuouswith respect to the strip pseudo-distance.

Lemma 4.31. Let x : S1 × [0, T∗) → Rn be a smooth curve shortening flow.Then for all 0 ≤ t1 ≤ t2 < T∗ there holds

D0(x(·, t1), x(·, t2)) ≤ C|t1 − t2|,

where the constant is the total absolute curvature C :=∫S1 |k|dµt1 .

Proof. We simply estimate the strip pseudo-distance by the area of the surfacewhich is generated by the flow F (u, t) = x(u, t) for (u, t) ∈ S1 × [t1, t2]. Sincedet(dxT dx) = k2|xu|2, the area is given by

Area(F ) =

∫ t2

t1

∫S1

|k|dµtdt.

The estimate follows since by lemma 3.4 the total absolute curvature is a non-increasing quantity.

103

Definition 4.32. Let f : R→ R be a Lipschitz function. The upper right Diniderivative of f(t) at t0 is defined by

D+t f(t0) := lim sup

h0

f(t0 + h)− f(t0)

h.

The strip pseudo-distance between two smooth curve shortening flows isnon-increasing.

Theorem 4.33. Let x, y : S1 × [0, T∗) → Rn be two smooth solutions of thecurve shortening flow. Then the strip pseudo-distance between the families ofcurves

f(t) := D0(x(·, t), y(·, t)),

is Lipschitz continuous and satisfies D+t f(t) ≤ 0 for t ∈ [0, T∗). Moreover, the

strip pseudo-distance f(t), t ∈ [0, T∗) is a non-increasing function.

Remark. The idea that area of a minimal surface spanned by a curve shorteningflow is a monotone quantity is from Altschuler and Grayson [8], Lemma 3.2.They actually prove that the disk-size satisfies D+

t D0(x(·, t)) ≤ −2π, wherex : S1× [0, T∗)→ R3 is a smooth curve shortening flow. Altschuler and Graysonuse this monotonicity to show that the limit of their ramp approximation, seeAppendix A, is unique.

In [49], Perelman used this idea in a more general way. He considers curveshortening flow x(·, t) : S1 → (M, g(t)), where (M, g(t)) is a compact Rieman-nian manifold that evolves by the Ricci flow. In this case the disk-size satisfiesthe growth rate estimate

D+t D0(x(·, t)) ≤ −2π − 1

2Rmin(t)D0(x(·, t)),

where Rmin(t) is minimum of the scalar curvature of the metric g(t), see Perel-man [49] Lemma 1.2. The strip pseudo-distance satisfies an analogously growthrate estimate

D+t D0(x(·, t), y(·, t)) ≤ (2n− 1) sup

M|Rm(t)|D0(x(·, t), y(·, t)),

where Rm(t) is the Riemann curvature of g(t), see Morgan and Tian [46] Lemma18.43 for more detail.

For convenience we first prove theorem 4.33 in the case where the curvesare embedded and disjoint. In this case we have disjoint Jordan curves andDouglas’ theorem B.3 ensures that the strip pseudo-distance is attained by aminimal surface.

Lemma 4.34. Theorem 4.33 holds under the additional assumption, that x(·, t)and y(·, t) are embedded and have disjoint images for all t ∈ [0, T∗).

Proof of lemma 4.34. By lemma 4.31 and the triangle inequality of the strippseudo-distance, we have the estimate

|f(t1)− f(t2)| ≤ D0(x(·, t1), x(·, t2)) +D0(y(·, t1), y(·, t2)) ≤ C|t1 − t2|.

As mention above, since solutions are embedded and disjoint, by the theoremB.3 of Douglas, at each time the strip pseudo-distance is attained by a minimal

104

surfaces F (·, t) ∈ C0(M,Rn)∩C∞(M,Rn), where M is either an annulus or thedisjoint union of two disks. Here we have to mention that in Douglas’ existencetheorem, F (·, t) is conformal. However, this would lead to a time dependentdomain. We simply re-parametrize the surfaces to a fixed domain M . Hence wehave

f(t) = D0(x(·, t), y(·, t)) = Area(F (·, t)).

Note that F (·, t) does not have to be continuous in time and can jump.For a fixed t0 ∈ [0, T∗), we vary F (·, t0) by a smooth family of diffeomor-

phisms φt : Rn → Rn, G(·, t) := φt(F (·, t0)) such that G(·, t0) = F (·, t0) onM and G(·, t) = F (·, t) on ∂M , i.e. φt0 = id and for Y := d/dt|t0φt we haveY (x(u, t0)) = k(x(u, t0)) resp. Y (y(u, t0)) = k(y(u, t0)). Since by lemma 4.31f is Lipschitz the upper right Dini derivative of f exists. By the minimality ofF (·, t) it follows

D+t f(t0) = lim sup

h0

f(t0 + h)− f(t0)

h≤ d

dt

∣∣∣∣t0

Area(G(·, t)).

Let ~n be the co-normal of F (·, 0) and kg the geodesic curvature of ∂F (·, t0). Notethat there holds kg = −〈k(x), ~n〉 and analogously for y. By the first variationformula of the area functional, we have

d

dt

∣∣∣∣t0

Area(G(·, t)) =

∫∂M

〈Y,~n〉+

∫M

〈H,Y 〉

=

∫S1

〈k(x), ~n〉+

∫S1

〈k(y), ~n〉

= −∫∂M

kg.

Now we use Gauss-Bonnet with interior branch points wi of order mi ≥ 1 andboundary branch points Wj of order Mj ≥ 1,∫

M

K +

∫∂M

kg = 2πχ(M) + 2π∑i

(mi − 1) + π∑j

(Mj − 1).

Since F (·, t0) is a minimal surface, the mean curvature vanishes H = λ1+λ2 = 0,hence the Gauss curvature K = λ1λ2 is non-positive. Since M is either anannulus or the union of two disks, the Euler characteristic is either χ(M) = 0or χ(M) = 2, respectively. Therefore, it follows that

D+t f(t0) ≤

∫M

K − 2πχ(M)− 2π∑i

(mi − 1)− π∑j

(Mj − 1) ≤ 0.

Hence, there holds D+t f(t0) ≤ 0 for t0 ∈ [0, T∗). Since the function f is Lips-

chitz, by Rademacher it is L1 almost everywhere differentiable and there holds

f(t1)− f(t2) ≤∫ t2

t1

D+t f(t)dt,

for all 0 ≤ t1 ≤ t2 < T∗.

105

In the general case where the curves x(·, t) and y(·, t) are neither injective nordisjoint at some time t ∈ [0, T∗), we use an injective and disjoint approximation.Such an approximation only exists if the co-dimension is high enough. Forthis reason we embed the flows in Rn+k. This is possible since an isometricembedding does not change the strip pseudo-distance.

Lemma 4.35. Let x, y ∈ C0,1(S1,Rn) and I : Rn → Rn+k be an isometricembedding. Then there holds D0(x, y) = D0(I x, I y).

Proof. To prove the inequality D0(Ix, Iy) ≤ D0(x, y), let F : S1×[0, 1]→ Rnbe an admissible map for D0(x, y) as in definition 4.2. Then I F is admissiblemap for D0(I x, I y) and the inequality follows since Area(F ) = Area(I F ).

To prove the opposite inequality D0(x, y) ≤ D0(I x, I y), let F be anadmissible map for D0(I x, I y). Modulo an isometry φ of Rn and an isometryψ of Rn+k any isometric embedding I : Rn → Rn+k is given by I(p) = (p, 0), i.e.I = ψ I φ. Let π : Rn+k → Rn be the projection π(p, q) = p. The inequalityfollows since F := ψ π ψ−1 F is an admissible map for D0(x, y), and thereholds Area(F ) ≤ Area(F ).

The injective and disjoint approximation exists only for a short time intervaland will follow from the following theorem which is proven in the appendix C.

Theorem 4.36. Let M be manifold without boundary, and let f : M → Rn bean immersion. Assume that there exist finitely many open pre-compact subsetsVk ⊂ M , 1 ≤ k ≤ N such that their closures are disjoint, i.e. V k ∩ V l = ∅ fork 6= l, and such that for all x, y ∈ M with x 6= y and f(x) = f(y) there existk 6= l with x ∈ Vk and y ∈ Vl. Then there exist ε > 0 and smooth cutoff functionsφk : M → [0, 1] with supp(φk) ⊂ Vk such that for every v1 ∈ Bnε (0) there existsa set of Ln-measure zero N2 = N2(v1) ⊂ Rn, for every v2 ∈ Bnε (0) \ N2 thereexists a set of Ln-measure zero N3 = N3(v1, v2), and so on, there exists a set ofLn-measure zero NN = NN (v1, ..., vN−1) such that for every vN ∈ Bnε (0) \NNsuch that the map

f(x) +

N∑k=1

φk(x)vk,

only has transversal self-intersections.

Proof of theorem 4.33. Fix two curve shortening flows x, y : S1 × [0, T∗)→ Rn.Claim 1. For every t0 ∈ (0, T∗) there exist an open neighborhood J ⊂ (0, T∗)

of t0 and smooth maps xi, yi ∈ C∞(S1× (t0, t0 +τ),Rn+k) such that xi(·, t) andyi(·, t) are embeddings with disjoint images for all t ∈ (t0, t0 + τ), and

xi → I x, yi → I y in C∞(S1 × J),

where I : Rn → Rn+k, I(p) := (p, 0) and k := max1, 5− n.Proof. Fix t0 ∈ (0, T∗). We first only show that there exist an open neigh-

borhood J ⊂ (0, T∗) of t0 and smooth maps xi : S1 × J → Rn+k such thatxi(·, t) is an embedding and xi smoothly converges to I x on S1× (0, T∗ − t0).

Since x is a smooth curve shortening flow, the maps x(·, t) are immersionsfor all t ∈ [0, T∗). Clearly, x on S1× (0, T∗) is not an immersion if the curvaturevanishes somewhere. This is the reason why we only can apply theorem 4.36 toa modified space-time map. We define the map X : S1 × [0, T∗)→ Rn+k by

X(u, t) := I(x(u, t)) + (0, t) ∈ Rn+k.

106

For n ≥ 4, this is the usual space-time map X(u, t) = (x(u, t), t) ∈ Rn+1, and forn ≤ 3 it is X(u, t) = (x(u, t), 0, t) ∈ R5. By construction there holds n+ k ≥ 5.Note that X is an immersion on S1×(0, T∗) if and only if x(·, t) is an immersionfor all t ∈ (0, T∗). Hence X is an immersion.

Let B = (u, t) : ∃(v, s) 6= (u, t), X(u, t) = X(v, s) be the set where X isnot injective. Since by definition X(u, t) = X(v, s) implies t = s, there holdsB ∩ (S1 × t) = B(t)× t, where

B(t) := u ∈ S1 : ∃v 6= u, x(u, t) = x(v, t).

Note that B(t) is finite for all t ∈ (0, T∗), since for each positive time t ∈ (0, T∗)the smooth curve shortening flow x(·, t) is analytic up to a diffeomorphism.

Since B(t0) is finite, there exist open subsets Uk ⊂ S1, 1 ≤ k ≤ N withdisjoint closure such that each Uk contains exactly one point of B(t0).

We claim that there exists an open neighborhood J ⊂ (0, T∗) of t0 suchthat Vk = Uk × J satisfy the condition of theorem 4.36 for the map f = Xon S1 × J . Proof by contradiction. Otherwise there exist ti → t0 and ui 6= viwith x(ui, ti) = x(vi, ti), and for every i ≥ 1 there do not exist k 6= l with(ui, ti) ∈ Vk and (vi, ti) ∈ Vl. By compactness we can assume that ui → u andvi → v, and (ui, ti) ∈ Vk and (vi, ti) ∈ Vl for some k, l. But this implies k = land hence (ui, ti) ∈ Vk for some k. Since x(ui, ti) = x(vi, ti) in the limit impliesx(u, t0) = x(v, t0), by definition of Uk there holds u = v. But this leads to

0 =x(ui, ti)− x(vi, ti)

ui − vi→ ∂

∂ux(u, t0),

which is a contradiction since x(·, t0) is an immersion.Hence, we can apply theorem 4.36 to X : S1 × J → Rn+k, to get the

approximation Xi : S1× J → Rn+k which converges in C∞ to X and such thatthe Xi only have transversal self-intersection points. Analogously, we find anapproximation Yi(u, t)→ Y := I(y(u, t) + (0, t) on S1 × J . By the intersectionof the open neighborhoods of t0, we may assume that this holds for the sameJ . Moreover, by corollary C.3 we can translate Yi by a vector vi ∈ Rn+k withvi → 0 such that all intersection points of Xi with Yi+vi are transversal. Hencewe have fund maps Xi, Yi which smoothly converge to X, Y and such that forall i ≥ 1 all self-intersection points of Xi and Yi as well as all intersection pointsof Xi with Yi are transversal. Since n + k ≥ 5, there are no transversal self-intersection points. Hence the maps Xi, Yi are injective and has disjoint images.In particular, the maps xi, yi : S1 × J → Rn+k defined by

xi(u, t) := Xi(u, t)− (0, t) and yi(u, t) := Yi(u, t)− (0, t),

satisfy for all t ∈ J that xi(·, t) and yi(·, t) are injective and have disjoint images.By construction xi and yi smoothly converge to I x and I y, respectively.

Hence for i big enough, xi(·, t) and yi(·, t) are immersions for all t ∈ J . There-fore, xi(·, t) and yi(·, t) are embeddings with disjoint images for all t ∈ J . Thisproves our claim 1.

For t ∈ J we define the functions

fi(t) := D0(xi(·, t), yi(·, t)).

107

Claim 2. There exist i0 and C such that Lip(fi) ≤ C for all i ≥ i0.Proof. By the triangle inequality of the strip pseudo-distance, we have theestimate

|fi(t1)− fi(t2)| ≤ D0(xi(·, t1), xi(·, t2)) +D0(yi(·, t1), yi(·, t2)).

Since xi resp. yi converge smoothly to I x resp. I y, by corollary 4.20 andlemma 4.35 there exists i0 such that for all i ≥ i0 there holds

D0(xi(·, t1), xi(·, t2)) ≤ 2D0(x(·, t1), x(·, t2)),

and an analogous estimate holds for y. By lemma 4.31 this implies the inequality

|fi(t1)− fi(t2)| ≤ 2C|t1 − t2|,

for all i ≥ i0, where C is the sum of total absolute curvatures of the curves xand y at time t1.

Claim 3. There holds fi → f in C0(J,R).Proof. The claim follows by lemma 4.35 and lemma 4.20 via the estimate

|fi(t)− f(t)| ≤ C (‖xi(·, t)− I x(·, t))‖C0 + ‖yi(·, t)− I y(·, t))‖C0) .

From now, the proof is similar Proof of lemma 4.34. But since xi and yionly approximate the smooth curve shortening flows I x and I y, we can notuse lemma 4.34 directly.

Since the images of xi(·, t) and yi(·, t) are disjoint Jordan curves, and bytheorem B.3 the strip pseudo-distance is attained by some minimal surfacesFi(·, t) ∈ C0(Mi,Rn) ∩ C∞(Mi,Rn), i.e. fi(t) = Area(Fi(·, t)) for t ∈ J , whereeither Mi = S1 × [0, 1] or Mi = B1(0)× 0, 1.

We fix t1 ∈ J and vary the minimal surface Fi(·, t1) by a smooth fam-ily of diffeomorphisms φi(·, t) : Rn+k → Rn+k such that φi(·, t1) = id andφi(xi(u, t1), t) = xi(u, t) resp. φi(yi(u, t1), t) = yi(u, t). Hence, the generatingvector fields Yi := d/dt|t1φi(·, t) satisfy Yi(xi(u, t1)) = d/dt|t1xi(u, t), and analo-gously for yi. By claim 2 the functions fi are Lipschitz continuous and thereforeD+t fi exists. By the minimality of fi(t) = Area(Fi(·, t)) it follows that

D+t fi(t1) = lim sup

h0

fi(t1 + h)− fi(t1)

h≤ d

dt

∣∣∣∣t1

Area(φi(Fi(·, t1), t)).

Let ~ni be the co-normal of Fi(·, t1) and kig the geodesic curvature and ki be

the mean curvature of ∂Fi(·, t1). There holds 〈ki, ~ni〉 = −kig and since Fi(·, t1)is minimal its the mean curvature Hi vanishes on Mi. Together with the firstvariation formula of the area functional, it follows that

d

dt

∣∣∣∣t1

Area(φi(Fi(·, t1), t)) =

∫∂Mi

〈Yi, ~ni〉dµi +

∫Mi

〈Hi, Yi〉

=

∫∂Mi

〈Yi − ki, ~ni〉dµi −∫∂Mi

kigdµi

As in lemma 4.34 since Fi(·, t1) is minimal and the Euler characteristic satisfiesχ(Mi) ≤ 2, by Gauss-Bonnet’s theorem it follows that

D+t f(t1) ≤

∫∂M

|Yi − ki|dµi. (122)

108

Since xi(·, t1)→ I x(·, t1) there holds

|Yi(xi(u, t1))− k(xi(u, t1))| = |Dtxi(u, t1)− k(xi(u, t1))|→ |DtI x(u, t1)− I k(x(u, t1))| = 0,

and analogously for y. Hence the first term on the right hand side of (122)converges to zero, and there holds D+

t fi(t1) ≤ δi(t1), where

δi(t) := max

supu∈S1

|Dtxi(u, t)− k(xi(u, t))|, supu∈S1

|Dtyi(u, t)− k(yi(u, t))|.

Since xi → I x and yi → I y in C∞(S1 × J), for every compact set J ⊂ Jthere holds

supt∈J

δi(t)→ 0, (i→∞).

By claim 3 this implies that f is non-increasing in J . Together with claim 1 thisshows that f is non-increasing in [0, T∗). In particular, this implies D+

t f(t) ≤ 0for t ∈ [0, T∗).

4.3 A Lower Area Bound

As an introduction, we consider a convex closed embedded curve x ∈ C2(S1,R2)with bounded curvature |k| ≤M . Since x is convex, for each point on x(S1), adisk of radius 1/M tangential to the curve, lies entirely in the region which thecurves bounds. The area of the bounded region corresponds to the definition ofD0(x), and this shows the lower bound

D0(x) ≥ π

M2.

We will generalize this lower bound in the following three ways.

1. To closed immersed curves in Rn with bounded curvature.

2. To immersed arcs in Rn, where the lower bound holds for D(x)2 instead ofD0(x).

3. To Lipschitz ε−graphs with Lipε(x) ≤ 1, where the inequality holds with εin place of 1/M .

First we prove a lower bound for D(x) for curves x that satisfy a local lowerbound of the extrinsic distance in terms of the intrinsic distance. All otherestimates will follow from this lower bound.

Lemma 4.37. Let x ∈ C0,1([0, 1],Rn) be a Lipschitz curve. Assume that thereexist ε > 0 and λ ∈ [λ0, 1], where λ0 := (2π)−1, such that for all u1, u2 ∈ [0, 1]with distx(u1, u2) ≤ ε there holds

|x(u1)− x(u2)| ≥ λ distx(u1, u2). (123)

Then the disk-size of x is bounded from below by

D(x) ≥ c0λminε, L(x),

where c0 := 2−3π−1/2.

109

Remark. Note that since λ ≥ (2π)−1 there holds

D(x) ≥ c1 minε, L(x),

where c1 := 2−4π−3/2.

Proof. Fix x ∈ C0,1([0, 1],Rn). Recall the definition of the disk-size

D(x) = infF∈A

√A(F ) +

L(v)√4π,

whereA consists of all Lipschitz maps F : B1 → Rn such that the boundary mapsatisfies ∂F φ = x ? v, where φ(u) := exp(2πiu), u ∈ [0, 1] and v : [0, 1]→ Rnis a Lipschitz curve. By an approximation of F in W 1,2, for every η > 0 thereexists a Lipschitz map F : B1 → Rn that is smooth in the interior of B1 suchthat

D(x) ≤√A(F ) +

L(v)√4π≤ D(x) + η, (124)

where x = y φ, y := ∂F |S−, v := ∂F |S+, and S+, S− denote the upper andlower hemisphere of ∂B1.

We define

F(F ) :=√A(F ) +

L(v)√4π. (125)

Claim.F(F ) ≥ c0λminε, L(y). (126)

Note that by (124) the claim implies the lemma in the limit η → 0.Since F is smooth in the interior of B1 we can define the pullback metric

(g)kl = DkFDlF and since F is Lipschitz on ∂B1 this introduces the distance

distF (p, q) := inf

∫ 1

0

|Dtγ(t)|gdt : γ ∈ C∞([0, 1], B1), p = γ(0), q = γ(1)

,

p, q ∈ B1. We denote by disty(u, v), u, v ∈ S− the distance on the curve y,which is the length L(y|[u, v]). Clearly there always holds

|F (u)− F (v)|Rn ≤ distF (u, v) ≤ disty(u, v), u, v ∈ S−. (127)

Proof of the claim. In case L(y) ≤ ε, the claim follows by (125) and (123)via

F(F ) ≥ L(v)√4π≥ |y(0)− y(1)|√

4π≥ λ√

4πL(y) = 4c0λL(y).

In case L(y) > ε, we argue by contradiction, and assume

F(F ) < c0λε =λε

4√

4π. (128)

We will inductively define a nested sequence of intervals Ik := [ak, bk] ⊃ Ik+1,k = 0, ...,m+ 1, where I0 := S−, such that

Lk := L(y|Ik) > ε, k = 0, ...,m (129)

dk := distF (ak, bk) <λε

4, k = 0, ...,m+ 1. (130)

110

We stop the procedure when Lm+1 ≤ ε, and this condition will be satisfiedin finitely many steps. Note that for k = 0 condition (129) is equivalent toL(y) > ε, and (130) follows by (128) via

d0 ≤ L(v) ≤√

4πF(F ) <λε

4.

Assume I0, ..., Ik have already been constructed satisfying (129) and (130). Weaim to define the interval Ik+1. We define

∆k := supu∈Ik

min distF (u, ak), distF (u, bk) .

By compactness, the exist uk ∈ Ik, where this maximum is attained, and wemay assume

distF (ak, uk) ≥ ∆k = distF (uk, bk). (131)

By definition of ∆k, we have y(Ik) ⊂ B∆k(y(ak)) ∪ B∆k

(y(bk)), and since theEuclidean distance between the center points is bounded by dk, it follows

y(Ik) ⊂ B∆k+dk/2(pk), (132)

where pk := (y(ak) + y(bk))/2.By (129) the curve segment y|Ik has length bigger than ε. In particular,

there exist u1, u2 ∈ Ik such that L(y|[u1, u2]) = ε and by the assumed property(123) on the curve x = y φ this implies |y(u1)− y(u2)| ≥ λε. This leads to thelower bound ∆k + dk/2 ≥ λε/2, for the radius given in (132). Together with(130) we have

2∆k + dk ≥ λε > 4dk,

and this implies the estimates

∆k >3

2dk, (133)

∆k >3

8λε. (134)

By the co-area formula with the function fk(z) := distF (uk, z), z ∈ B1 weobtain ∫ ∆k

0

LengthF (fk = l)dl ≤ Area(F ) ≤ F(F )2.

Therefore, there exists δk ∈ [∆k/3, 2∆k/3] such that

LengthF (fk = δk) ≤3F(F )2

∆k. (135)

Now, we use the fact that the surface is of disk type. The set fk = δkmay consist of many connected components, but since δk < ∆k either thereexists one component that separates uk from both ak and bk, or there exist twocomponents of fk = δk such that one separates ak from both uk and bk andthe other separates bk from both uk and ak. This two cases correspond to theleft- and right-hand side of figure 5.

More precisely, there exist ak+1 ∈ [ak, uk] and bk+1 ∈ [uk, bk] such thateither ak+1 and bk+1 are connected by a component of fk = δk, or there aretwo components which connect ak+1 resp. bk+1 to S1 \ [ak, bk].

111

ak

ak+1

bk

bk+1

uk

fk = δk ak

ak+1

bk

bk+1

uk

fk = δk

Figure 5: Disk-Type Surface

Now, we rule out the second case. If there exists a component of fk = δkthat separates ak from bk, then there exists a point p ∈ fk = δk such that

distF (p, ak, bk) ≤distF (ak, bk)

2=dk2.

By (131) and (133), this leads to the contradiction

δk = distF (uk, p) ≥ ∆k − distF (p, ak, bk) ≥ ∆k −dk2

>2∆k

3≥ δk.

Hence, we are in the first case. We now check that the condition (130) holds forIk+1 := [ak+1, bk+1]. By (128), (134) and (135) there holds

dk+1 = distF (ak+1, bk+1) ≤ LengthF (fk = δk) ≤3F(F )2

∆k<λε

8π. (136)

In particular, this proves that the condition (130) holds for dk+1, and we caniterate the procedure as long as condition (129) holds. Moreover, by constructionwe have

distF (ak, ak+1) ≥ distF (ak, uk)− distF (ak+1, uk) = ∆k − δk ≥∆k

3,

and analogously for bk+1, bk. This together with (127) and (134) implies

Lk − Lk+1 ≥2∆k

3≥ λε

4, (137)

where Lk+1 = L(y|Ik+1) and Ik+1 = [ak+1, bb+1]. Since the length decreases bya fixed amount, there exists m ≥ 1 such that Lm+1 ≤ ε.

For the interval Im+1 = [am+1, bm+1] we still have

distF (am+1, um) = δm ≥∆m

3,

and analogously for bm+1. This implies the lower length bound

Lm+1 ≥2∆m

3. (138)

112

Since Lm+1 ≤ ε, by assumption (123) and (127), it follows that dm+1 ≥ λLm+1.Together with inequalities (134), (138) and (136) this leads to

λ2ε

4≤ 2λ∆m

3≤ λLm+1 ≤ dm+1 <

λε

8π,

which contradicts the assumption λ ≥ λ0 = (2π)−1. This proves the claim (126)and the lemma.

Curves with bounded curvature as well as locally Lipschitz graph-like curvesof definition 3.33 satisfy a local lower bound of the extrinsic distance in terms ofthe intrinsic distance. The following two lemmata give quantitative estimatesof this fact.

Lemma 4.38. Let x ∈ C2([0, 1],Rn) be an immersed curve with bounded cur-vature |k| ≤M . Then for all u1, u2 ∈ [0, 1] there holds

|x(u1)− x(u2)| ≥(

1− M2distx(u1, u2)2

24

)distx(u1, u2). (139)

Proof. We may assume that x is parametrized by arc-length, and hence thereholds |u1 − u2| = distx(u1, u2). The extrinsic distance in Rn is bounded by

|x(u1)− x(u2)| =∣∣∣∣∫ u2

u1

xudu

∣∣∣∣ ≥ ∫ u2

u1

〈xu, v〉 du,

for every fixed vector v with |v| = 1. We choose v := xu(u∗) where u∗ :=(u1 + u2)/2. By the curvature bound, for all u ∈ [u1, u2] there holds

|xu(u)− v| ≤∫ u

u∗|xuu|du ≤M |u− u∗|.

Since |xu(u)| = |v| = 1 we have the relation

|xu(u)− v|2 = 2(1− 〈xu, v〉).

Together we the estimate above we have

|x(u1)− x(u2)| ≥∫ u2

u1

(1− |xu(u)− v|2

2

)du ≥

∫ u2

u1

(1− M2|u− u∗|2

2

)du.

By integration this implies the estimate

|x(u1)− x(u2)| ≥(

1− M2|u1 − u2|2

24

)|u1 − u2|.

Lemma 4.39. Let x : [0, 1] → Rn be a Lipschitz ε−graph, i.e. Lipε(x) < ∞.Then for all u1, u2 ∈ [0, 1] with distx(u1, u2) ≤ ε there holds

|x(u1)− x(u2)| ≥ distx(u1, u2)√1 + Lipε(x)2

. (140)

113

Proof. Assume distx(u1, u2) ≤ ε and let I := [u1, u2]. We may assume thatx is parametrized by arc-length, i.e. |xu| = 1 for almost every u ∈ [0, 1]. Bydefinition of Lipε(x), there exists a unit vector v such that

〈xu(u), v〉 ≥ cos(tan−1(Lipε(x))

)=

1√1 + Lipε(x)2

,

holds for almost every u ∈ I. Analogously to the case with bounded curvature,we conclude

|x(u1)− x(u2)| ≥∫I

〈xu, v〉 du ≥distx(u1, u2)√1 + Lipε(x)2

.

By lemma 4.37 and the two lemmata above, we have the following fourresults.

Corollary 4.40. Let x : [0, 1]→ Rn be a Lipschitz ε−graph, with Lipε(x) ≤ 1.Then the disk-size of x is bounded from below by

D(x) ≥ c1 minε, L(x),

where c1 := 2−4π−3/2.

Proof. By lemma 4.39 and the assumption Lipε(x) ≤ 1 we have

λ :=1√

1 + Lipε(x)2≥ λ0 = (2π)−1.

Hence the corollary follows by lemma 4.37 and its remark.

Corollary 4.41. Let x ∈ C2([0, 1],Rn) be an immersed curve, with boundedcurvature |k| ≤M . Then the disk-size of x is bounded from below by

D(x) ≥ c2 min4/M,L(x),

where c2 = 2−33−1π−1/2.

Proof. By lemma 4.38 the condition (123) holds for ε = 4/M > 0 and

λ = 1− 42

24=

1

3> λ0 = (2π)−1.

The corollary follows by lemma 4.37 with c2 = λc0.

Corollary 4.42. Let x : S1 → Rn be a Lipschitz ε−graph, with Lipε(x) ≤ 1.Then the disk-size of x is bounded from below by

D0(x) ≥ c3ε2,

where c3 := 2−8π−3.

Proof. By lemma 3.42 the length is bounded below by

L(x) ≥ πε

4 tan−1(1)= ε.

Hence by corollary 4.40 we have D0(x) = D(x)2 ≥ (c1ε)2 = c3ε

2.

114

Corollary 4.43. Let x ∈ C2(S1,Rn) be an immersed closed curve, with boundedcurvature |k| ≤M . Then the disk-size of x is bounded from below by

D0(x) ≥ c4M2

,

where c4 := 2−23−2π−1.

Remark. The estimate of the corollary does not seems to be optimal. We expectthat there holds D0(x) ≥ π/M2, as for curves in R2. Intuitively this is true bythe monotonicity of the area ratio for minimal surfaces, see lemma B.4.

Proof. For closed C2 curves in Rn by Fenchel’s inequality [24] there holds

2π ≤∫S1

|k|dµ.

Therefore, the length of the curve is bounded from below by L(x) ≥ 2π/M >4/M . Moreover, by lemma 4.6 there holds D0(x) = D(x)2. Hence by corollary4.41 we have

D0(x) ≥ c22(4/M)2 = c4/M2.

4.4 Local Lipschitz Graph Estimate

Lemma 4.44. Let x, y ∈ C0,1([0, 1],Rn) be Lipschitz ε−graphs with Lipε(x),Lipε(y) ≤ 1. Assume that the curves have small strip pseudo-distance

D(x, y) ≤ c1ε

2CN1, (141)

where c1 := 2−4π−3/2 and C1 := (1 + 1/(2π)) and N :=⌈(4π3/2/c1)(L(x)/ε)

⌉.

Then there exist 0 = a1 < a1 < ... < ap+1 = 1 and 0 = b1 < ... < bp+1 = 1,p ≤ 2N such that for all Ii := [ai, ai+1] and Ji := [bi, bi+1], i = 1, ..., p there hold

10−3ε ≤ L(x|Ii), L(y|Ji) ≤ ε,

andD(x|Ii, y|Ji) ≤ CN1 D(x, y).

Proof. By corollary 4.40 for every interval I, J ⊂ [0, 1] there hold

D(x|I) ≥ c1 minε, L(x|I),D(y|J) ≥ c1 minε, L(y|J). (142)

We assume that either L(x) > ε or L(y) > ε since otherwise we are done.The following procedure is symmetric in x and y. For simplicity we assume thatwe are in the first case L(x) > ε. Hence estimate (142) with I = [0, 1] impliesD(x) ≥ c1ε. Assumption (141) implies D(x, y) ≤ c1ε/2, and therefore thereholds

D(x)−D(x, y) ≥ c1ε/2 ≥ D(x, y). (143)

115

By lemma 4.24 there exist 0 < a, b < 1 such that for I := [0, a] and J := [0, b]there hold

D(x|I, y|J),D(x|Ic, y|Jc) ≤(

1 +D(x, y)

2π(D(x)−D(x, y))

)D(x, y)

and

L(x|I), L(x|Ic), L(y|J), L(y|Jc) ≥√π

2(D(x)−D(x, y)).

Together with estimate (143) this implies

D(x|I, y|J),D(x|Ic, y|Jc) ≤(

1 +1

2π

)D(x, y) = C1D(x, y), (144)

and

L(x|I), L(x|Ic), L(y|J), L(y|Jc) ≥√π

4c1ε ≥ 10−3ε. (145)

We use this procedure several times and split [0, 1] into intervals Ii and Ji suchthat L(x|Ii) < ε and L(y|Ji) < ε.

By inequality (142) and assumption (141) there holds D(x|I, y|J) ≤ c1ε/2for at least N steps. As long as either L(x|I) > ε or L(y|J) > ε, inequality (142)implies either D(x|I) ≥ c1ε or D(y|J) ≥ c1ε and we can iterate the procedurewith either x or y.

By inequality (145) at each step the length L(x|I) = L(x) − L(x|Ic) andL(y|J) = L(y) − L(y|Jc) are decreasing at least by a fixed amount, and N isdefine such that

N

√π

4c1ε =

⌈4L(x)

c1ε

⌉ √π

4c1ε ≥ minL(x), L(y).

Hence there holds 10−3ε ≤ L(x|I), L(y|J) ≤ ε in less than N steps.

Now, we improve the lemma 4.44 to closed curves and the strip pseudo-distance D0.

Corollary 4.45. Let x, y ∈ C0,1(S1,Rn) be closed Lipschitz ε−graphs withLipε(x), Lipε(y) ≤ 1 and small strip pseudo-distance√

D0(x, y) ≤ c1ε

4CN1, (146)

where c1 := 2−4π−3/2 and C1 := (1 + 1/(2π)) and N :=⌈(4π3/2/c1)(L(x)/ε)

⌉.

Then there exist 0 ≤ a, b ≤ 1, p ≤ 2N , 0 = a1 < a1 < ... < ap+1 = 1 and0 = b1 < ... < bp+1 = 1 such that for the intervals

Ii :=[e2πi(ai+a), e2πi(ai+1+a)

], Ji :=

[e2πi(bi+b), e2πi(bi+1+b)

],

there hold10−3ε ≤ L(x|Ii), L(y|Ji) ≤ ε,

andD(x|Ii, y|Ji) ≤ CN1

√D0(x, y).

116

Proof. We first use lemma 4.26 to get the estimate

D(x, y) ≤√D0(x, y) +

D0(x, y)

2π√D0(x)−D0(x, y)

,

for the curves x(u) = x(exp(2πi(u+ a)) and y(u) = y(exp(2πi(u+ b)).By corollary 4.42 and assumption there holds√

D0(x) ≥ (c1/2)ε ≥ 2√D0(x, y).

Together with the estimate above this shows

D(x, y) ≤ 2√D0(x, y).

Hence we can apply lemma 4.44 to get the result.

The following theorem can be compared with Lemma 5.3 from Altschulerand Grayson in [8]. It shows that if a Lipschitz ε−graph x and a Lipschitzη−graph y with η << ε are very close in the strip pseudo-distance, then y isalso a Lipschitz ε−graph and the curves are Lipschitz close.

Theorem 4.46. Let x, y ∈ C0,1(S1,Rn). Assume x is a Lipschitz ε−graphand y is a Lipschitz η−graph, with Lipη(y), Lipε(x) ≤ 1/5. Then there existsc = c(L(x), η, Lipε(x), Lipη(y)) > 0 such that if D0(x, y) ≤ c then

Lipε(y) ≤ 20Lipε(x) + 12Lipη(y). (147)

Moreover, if x, y are parametrized by constant speed, then there exists abijective, orientation preserving Lipschitz map f : S1 → S1 such that

f ′(u) ∈ [8/10, 12/10], a.e. u ∈ S1,

Lip(x− y f) ≤ 4(Lipε(x) + Lipη(y)),

|x− y f |C0 ≤ 4η(Lipε(x) + Lipη(y)).

Proof. Note that we can assume that η < ε. By corollary 4.45 there existconstants c(L(x)/η), C(L(x)/η) such that if D0(x, y) ≤ c(L(x)/η)η then thereexist p = p(L(x)/η) and partitions of S1 in intervals Ii, Ji, i = 1, ..p such that10−3η ≤ L(x|Ii), L(y|Ji) ≤ η and D(x|Ii, y|Ji) ≤ C(L(x)/η)

√D0(x, y).

Therefore, we have a small strip pseudo-distance between Lipschitz graphsx(Ii) and y(Ji). For the sake of simplicity, we write I = Ii and J = Ji.

Note that the tangent vectors T x, T y of both curves are nearly constant, i.e.for almost every u1, u2 ∈ I there holds

|T x(u1)− T x(u2)| ≤ sin(αε(x)) =2Lipε(x)

1 + Lipε(x)2≤ 2Lipε(x),

and analogously for y.Claim 1. For all u ∈ I and v ∈ J there holds

|T x(u)− T y(v)| ≤ 4(Lipε(x) + Lipη(y)).

117

Proof. Fix u∗ ∈ I and v∗ ∈ J . We denote the boundary points of theintervals and curves by [a1, a2] := I, [b1, b2] := J , xj = x(aj) and yj = y(bj).We estimate the difference between the tangent vectors by

|T x(u∗)− T y(v∗)| ≤∣∣∣∣T x(u∗)− x2 − x1

L(x|I)

∣∣∣∣+

∣∣∣∣x2 − x1

L(x|I)− y2 − y1

L(x|I)

∣∣∣∣+

∣∣∣∣y2 − y1

L(x|I)− y2 − y1

L(y|J)

∣∣∣∣+

∣∣∣∣T yx(v∗)− y2 − y1

L(y|J)

∣∣∣∣ .The first term on the right hand side is bounded by∣∣∣∣T x(u∗)− x2 − x1

L(x|I)

∣∣∣∣ ≤ 1

L(x|I)

∫I

|T x(u∗)− T x(u)| dµx(u) ≤ 2Lipε(x).

Analogously, the last term is bounded by 2Lipη(y).We choose D0(x, y) ≤ c small enough such that there holds

D(x|I, y|J)

L(x|I)≤C(L(x)/η)

√D0(x, y)

10−3η≤ 1

2minLipε(x), Lipη(y). (148)

By construction and this choice, the second term is bounded by

|x1 − y1|+ |x2 − y2|L(x|I)

≤ D(x|I, y|J)

L(x|I)≤ Lipε(x) + Lipη(y).

To get an estimate for third term we have to compare length of the segments

L(y|J) ≤√

1 + Lipη(y)2 |y2 − y1|

≤√

1 + Lipη(y)2 (|x2 − x1|+ |x2 − y2|+ |x1 − y1|)

≤√

1 + Lipη(y)2 (L(x|I) +D(x|I, y|J)) ,

≤√

1 + Lipη(y)2L(x|I) + Lipε(x)L(x|I),

L(y|J) ≥ |y2 − y1| ≥ |x2 − x1| − |x2 − y2| − |x1 − y1|

≥ L(x|I)√1 + Lipε(x)2

−D(x|I, y|J)

≥ L(x|I)√1 + Lipε(x)2

− Lipη(y)L(x|I).

The last two inequalities implies the following estimate of the third term∣∣∣∣y2 − y1

L(x|I)− y2 − y1

L(y|J)

∣∣∣∣ ≤ ∣∣∣∣L(y|J)− L(x|I)

L(x|I)

∣∣∣∣ ≤ Lipε(x) + Lipη(y).

Together this shows the estimate

|T x(u∗)− T y(v∗)| ≤ 4(Lipε(x) + Lipη(y)),

and proves claim 1.

118

Claim 2. y is a Lipschitz ε−graph and satisfies (147).Proof. By claim 1, the segments x|I and y|J has nearly constant tangent vectorsin the same direction and there boundary points are very close. In particular,for length of these segments there holds L(y|I) ∈ (0.8, 1.2)L(x|I).

Let W ⊂ S1 such that the segment y|W has length ε. We may assume thatthe set W is covered by interval Ji, i = 1, ..., p. Let V := ∪pi=1Ii be the union ofthe corresponding interval of the curve x. By the length estimate of the smallsegments, there holds L(x|W ) ≤ 2ε.

Let β be the maximal angle between tangent vectors of x|I and y|J . By theclaim above there holds tan (β) ≤ 4(Lipε(x) + Lipη(y)). The maximal anglebetween two tangent vectors of y|W is bounded by 2αε(x) + 2β. This leads tothe estimate

Lipε(y) ≤ tan (2αε(x) + 2β)

= ≤ 20Lipε(x) + 12Lipη(y),

and proves claim 2.Claim. If x, y are parametrized proportional to arc-length, then there are

Lipschitz close modulo a bijective Lipschitz map f .Proof. We define f as the piecewise affine map, mapping Ii to Ji, i.e. foru ∈ Ii = [ai, ai+1] we define f by

f(u) := bi + (u− ai)bi+1 − biai+1 − ai

∈ Ji = [bi, bi+1].

By the length estimate of the segments there holds f ′ ∈ [0.8, 1.2]. By claim 1,for the Lipschitz semi-norm we have

Lip(x− y f) ≤ 4(Lipε(x) + Lipη(y)).

By integration of this inequality and the estimate (148) there holds

|x− y f |C0 ≤ C(L(x)/η)√D0(x, y) + 2η(Lipε(x) + Lipη(y))

≤ 3η(Lipε(x) + Lipη(y)).

119

5 Weak Solution

In this chapter we will prove our main results existence, uniqueness and partialregularity of a weak solution.

In the first section we define a weak solution on time interval [0, T ] as a pair(x, U) consisting of an open set U that is dense in [0, T ] and a family of Lipschitzcurves x(·, t) for times t ∈ U . Compared to the definition in the introduction,for a technical reason, here we put the condition that U is a full measure setof [0, T ] into a separate item. Generally, a weak solution can stop at any timeT , but we will define the extinction time until a weak solution can be extendedand for which we will prove existence.

A weak solution is by definition modulo a locally in U time independent, non-decreasing degree one map of S1 equal to a smooth solution. We call such anon-decreasing Lipschitz map that is locally constant in U , a crashing map. Weshow that the crashing map and the corresponding smooth flow are unique upto an orientation preserving diffeomorphism. We define regular and past-regularpoint in the same spirit. In a neighborhood of a regular point, a weak solutionagrees modulo a crashing map with a smooth flow. At a past-regular pointthe same holds for a past neighborhood. We also define equivalence classes viacrashing maps, in which a weak solutions will be unique. We show that the openset U of a weak solution only dependents on the equivalence class. Moreover,we prove that the set U , the set of regular times and the set of past-regulartimes coincide on (0, T ).

In the second section we prove our uniqueness result. This will follow fromthis fact, that the strip pseudo-distance between two weak solutions is a non-increasing function of t ∈ U . Here it is essential that U is a full measure set.

In the third section we start with our existence proof. For a sequence ofsmooth solutions which converges at the initial time, we prove sub-sequentialconvergence on a open dense set U of [0, T ]. Here we use the estimates fromsection 4.4 and argue similar to the ramp convergence of Altschuler and Graysonin [8]. Moreover, we show that the limit satisfies all conditions of a weak solutionexcept that U is a full measure set.

In section 5.4 we extend the limit solution of the third section from U to[0, T ]. This is necessary to define past-singular and singular space-time points.Using the local regularity theorem from section 3.10, we prove that the parabolic,one dimensional Hausdorff measure of the past-singular space-time points isbounded. This implies that the Hausdorff dimension of the set of singular timesis less or equal than 1/2, and in particular shows that U is a full measure set.

In section 5.5, we prove that for every C1 immersion there exists a maximalweak solution with this initial condition. For this reason we define a waveapproximation that is an alternative to the ramp approximation given in [8].For such an approximation, the long time existence result from section 3.9,implies that the weak solution exists up the extinction time.

In the last section, we show that for a weak solution that arises as limitof smooth solutions the set of past-singular space points at every time slice isfinite, and their number is bounded by a multiple of the left limit of the totalabsolute curvature. This manly follows by the estimates for graph flows fromsection 3.6, and improves a result from Angenent in [6].

121

5.1 Definition and Properties

Definition 5.1. A pair (x, U) consisting of an open dense set U of (0, T ) anda family of Lipschitz curves x(·, t) : S1 → Rn for t ∈ U , is a weak solution tothe curve shortening flow on S1 × [0, T ] if

a) the family of Radon measures µt on S1 defined by

µt(ψ) :=

∫S1

ψ(u)|xu(u, t)|du, ψ ∈ C0(S1,R)

is non-increasing in U , i.e. for all t1, t2 ∈ U with t1 ≤ t2 there holds

µt1 ≥ µt2 , (149)

in the sense of Radon measures.

b) the family x(·, t), t ∈ U , is Holder 1/2 in U and Lipschitz continuous awayfrom zero with respect to the strip pseudo-distance, i.e. there exists a con-stant C <∞ such that for all t1, t2 ∈ U there holds

D0(x(·, t1), x(·, t2)) ≤ C|t1 − t2|1/2, (150)

and for all t0 > 0 there exists C(t0) <∞ such that for all t1, t2 ∈ U ∩ [t0, T ]there holds

D0(x(·, t1), x(·, t2)) ≤ C(t0)|t1 − t2|. (151)

c) there exists a map x : S1×U → Rn such that for every connected componentV of U there exists a non-decreasing, degree one Lipschitz map φV : S1 → S1

such thatx(u, t) = x(φV (u), t), (u, t) ∈ S1 × V, (152)

and the restriction x|S1 × V is a smooth curve shortening flow that can notbe continued beyond the interval V , if supV < T .

d) U is a full measure set of [0, T ].

To ensure that our solution extends the flow until it disappears, we will provethe existence of a maximal weak solution in the sense of the following definition.

Definition 5.2. A weak solution (x, U) on S1 × [0, T ∗] is called maximal, ifthere holds

limt→T∗t∈U

D0(x(·, t)) = 0,

and T ∗ is called the extinction time.

For a weak solution (x, U) the map x : S1 × U → Rn is not asserted tohave a continuous extension on S1× [0, T ]. The the monotonicity of the Radonmeasures together with the continuity in the strip pseudo-distance compensatesthis in a weak way, but this is not strong enough to connect the family uniquelyas maps. We will prove later that weak solutions are unique in the followingsense.

122

Definition 5.3. Two weak solutions (x1, U1) and (x2, U2) on S1 × [0, T ] arecalled equivalent, if for every open interval V ⊂ U1 ∩ U2 there exist non-decreasing, degree one Lipschitz maps φV , ψV : S1 → S1, and a smooth curveshortening flow xV : S1 × V → Rn such that

x1(u, t) = xV (φV (u), t), (u, t) ∈ S1 × V,x2(u, t) = xV (ψV (u), t), (u, t) ∈ S1 × V.

The non-decreasing, degree one Lipschitz map φ and the smooth curve short-ening flow x are unique up to an orientation preserving diffeomorphism.

Lemma 5.4. Let H,H1, H2 be either open, precompact intervals or H = H1 =H2 = S1, x1 : H1 → Rn, x2 : H2 → Rn smooth immersions, and φ1 : H → H1,φ2 : H → H2 either continuous, surjective, non-decreasing maps of intervals orcontinuous, non-decreasing, degree one maps of S1. Assume that there holdsx1 φ1 = x2 φ2. Then there exists an orientation preserving diffeomorphismf : H1 → H2 such that φ2 = f φ1 and x2 f = x1.

Proof. Since φ1, φ2 are either continuous, surjective, non-decreasing maps ofprecompact sets or continuous, non-decreasing, degree one maps of S1, for everyu ∈ H1 the set φ−1

1 (u) ⊂ H1 is a compact interval and the image φ2(φ−11 (u)) ⊂

H2 is again a compact interval. By assumption there holds

x1(u) = x1(φ1(φ−11 (u))) = x2(φ2(φ−1

1 (u))),

as sets. Since x2 is an immersion it is locally injective. Hence the closed intervalφ2(φ−1

1 (u)) consists of only one point and therefore we may define the functionf : H1 → H2 by f(u) := φ2(φ−1

1 (u)). By definition these function satisfiesx1 = x2f and fφ1 = φ2φ−1

1 φ1 = φ2. Analogously, there exists g : H2 → H1

such that g φ2 = φ1. We obtain f g φ2 = φ2 and g f φ1 = φ1 and sinceφ1, φ2 are surjective, there holds f g = id = g f . Hence f is bijective andsince x1, x2 are smooth immersions f and g are also smooth immersions. Sinceφ1, φ2 are of degree one, the diffeomorphism f is orientation preserving.

As a consequence of lemma 5.4, the open full measure set U of a weaksolution only depends on the equivalence class.

Lemma 5.5. Let (x1, U1) and (x2, U2) be two equivalent weak solutions. Thenthere holds U1 = U2.

Proof. By contradiction we assume that there exists t0 ∈ U1 \ U2. Let V1 bethe connected component in U1 which contains t0. Since U2 is a full measureset and in particular is dense in [0, T ] there holds

U2 ∩ V1 ∩ [0, t0] 6= ∅.

Hence there exists a connected component V2 ⊂ [0, t0] of U2 such that V :=V1 ∩ V2 6= ∅. Since the weak solutions are equivalent and by the definition5.1.c), there exist non-decreasing, degree one Lipschitz maps φV , ψV , φV1

, φV2:

S1 → S1, and smooth curve shortening flow xV , xV1, xV2

such that on S1 × Vthere hold

xV (φV (u), t) = x1(u, t) = xV1(φV1(u), t),

xV (ψV (u), t) = x2(u, t) = xV2(φV2(u), t).

123

By lemma 5.4 on S1×V the flows xV , xV1, xV2

are equal up to a time independentdiffeomorphism. Therefore, xV2

can be extended beyond the interval V2 up totime supV1. This contradicts 5.1.c) and thus the assumption that (x2, U2) is aweak solution.

Definition 5.6. Let U be an open set dense in (0, T ) and x(·, t) : S1 → Rn,t ∈ U a family of Lipschitz curves. A point (u0, t0) ∈ S1 × [0, T ] is called aregular respectively past-regular point of (x, U), if there exist δ > 0 such thatfor J := (t0 − δ, t0 + δ) respectively J := (t0 − δ, t0] there holds J ⊂ [0, T ],an open set H ⊂ S1 with u0 ∈ H, a non-decreasing, surjective Lipschitz mapφ : H → (0, 1), and a smooth curve shortening flow x : (0, 1) × J → Rn, suchthat

x(u, t) = x(φ(u), t), (u, t) ∈ H × (J ∩ U). (153)

We denote the set of regular points by R ⊂ S1 × [0, T ], the set of past-regular points by Rp ⊂ S1 × [0, T ], and define the set of singular points byS := S1× [0, T ] \R and the set of past-singular points by Sp := S1× [0, T ] \Rp.

Moreover, we call the time projection S := π(S) the set of singular times, andits complement R := [0, T ]\S the set of regular times. Similarly, Sp := π(Sp) isthe set of past-singular times, and Rp := [0, T ]\Sp the set of past-regular times,where π : S1 × R→ R is the canonical projection to time.

Generally there hold R ⊂ Rp and Sp ⊂ S. At a past-regular point (u0, t0)a weak solution agrees modulo a crashing map φ with smooth curve shorteningflow x : [0, 1]×(t0−δ, t0]→ Rn. Note that by definition 3.1 of the smooth flow fora semi-open time interval, the map x(·, t0) is a smooth immersion. Equivalently,all points v ∈ [0, 1] with φ(u0) = v are regular points in the sense of definition3.50 of the flow x : (0, 1) × (t0 − δ, t0) → Rn. In particular, at a past-regularpoint (u0, t0), one can smoothly continue the flow x beyond the time t0.

One reason why we have to distinguish between regular and past-regularpoints are hairs. Locally a hair, if one does not see a tip, is maybe smooth.Hence many points of a hair can be past-regular. As mention above, at a past-regular point, we can extend x smoothly into the future and via φ this extendsx locally. But a hair disappears instantly and therefore this local extension doesnot coincide with the weak solution. In fact, no point of a hair is regular.

The following lemma gives an alternative definition of regular and past-regular times.

Lemma 5.7. Let U be an open set dense in (0, T ) and x(·, t) : S1 → Rn,t ∈ U be a family of Lipschitz curves. For every regular time t0 ∈ R respectivelypast-regular time t0 ∈ Rp of (x, U) there exists δ > 0 such that for J := (t0 −δ, t0 + δ) respectively J := (t0 − δ, t0] there holds J ⊂ [0, T ] and there exist anon-decreasing, degree one, Lipschitz map φ : S1 → S1 and a smooth curveshortening flow x : S1 × J → Rn such that

x(u, t) = x(φ(u), t), (u, t) ∈ S1 × (J ∩ U).

Proof. Let t0 ∈ R. By definition ofR and since S1 is compact there exist a coverS1 = ∪Ni=1Hi of open intervals Hi, open connected neighborhoods Ji ⊂ [0, T ] oft0, non-decreasing Lipschitz maps φi : Hi → (0, 1), and smooth curve shorteningflows xi : (0, 1)× Ji → Rn, such that

x(u, t) = xi(φi(u), t), (u, t) ∈ Hi × Ji ∩ U.

124

The set J := ∩Ni=1Ji is again an open neighborhood of t0. We may assume thatthe open intervals Hi are ordered such that Hi ∩Hi+1 6= ∅, where HN+1 := H1.By lemma 5.4 there exists a diffeomorphism fi : φi(Vi) → φi+1(Vi) such thatφi+1(u) = fi(φi(u)) and xi+1(fi(u), t) = xi(u, t) holds for all u ∈ φi(Vi).

This allows us to glue the maps φi resp. xi together to maps φ : S1 → S1

resp. x : S1 × J → Rn. We define the space

V :=

N∐i=1

Im(φi)/ ∼,

where we identify fi(φi(u)) ∼ φi+1(u) for all u ∈ Hi ∩Hi+1. We define the mapφ : S1 → V by

φ(u) := φi(u), u ∈ Hi \Hi+1.

Note that V is diffeomorphic to S1 and φ is well define. Since φi are non-decreasing Lipschitz maps and fi are orientation persevering, φ is a non-decreasing,degree one map and there holds

Lip(φ) ≤ max1≤i≤N

Lip(φi) max1, Lip(fi).

The map x : V × J → Rn

x(v, t) := xi(v, t), v ∈ φi(Hi)

is well define, satisfies the smooth curve shortening flow equation and thereholds

x(u, t) = x(φ(u), t), (u, t) ∈ S1 × (J ∩ U).

This proves the lemma in the case of R and the proof for Rp is analogously.

We already have shown that the set open full measure set U of a weaksolution only depends on the equivalence class. Now we prove that U is actuallythe set of regular times.

Lemma 5.8. Let U be an open set dense in (0, T ) and x(·, t) : S1 → Rn, t ∈ Ube a family of Lipschitz curves. Assume (x, U) satisfies condition 5.1.c) of aweak solution. Then for the regular and past-regular times of (x, U) there holds

U = R = Rp ∩ [0, T ).

Remark. Generally, the stop time T could be past-regular. In this case the weaksolution can be extended and T will be a regular time of the extension. If (x, U)is a maximal weak solution then there holds equality U = R = Rp.

Proof. By lemma 5.7 there hold the inclusions U ⊂ R ⊂ Rp ∩ [0, T ).Let t0 ∈ Rp ∩ [0, T ). By lemma 5.7 there exists J = (t0 − δ, t0], a non-

decreasing, degree one Lipschitz map φ : S1 → S1, and a smooth curve short-ening flow x : S1 × J → Rn, such that

x(u, t) = x(φ(u), t), (u, t) ∈ S1 × (J ∩ U).

In particular, x(·, t0) is a smooth immersion and hence we can continue the flowsmoothly beyond the time t0.

125

Since U is dense, there exists a connected component V of U such thatV ∩J is not empty. By condition 5.1.c) there exist a non-decreasing, degree oneLipschitz map φ : S1 → S1 and a smooth solution xV : S1 × V such that

x(u, t) = xV (φV (u), t), (u, t) ∈ S1 × V.

Moreover, either xV is a maximal smooth solution or there holds T = supV .In the second case t0 < T implies t0 ∈ V ⊂ U and we are done. In the firstcase, by lemma 5.4 xV and x coincide on V ∩ J up to a orientation preservingdiffeomorphism. Since x can be continue beyond t0, this also implies t0 ∈ V .

5.2 Uniqueness

Theorem 5.9. Let (x, Ux), (y, Uy) be weak solutions on S1 × [0, T ] and U :=Ux ∩ Uy. Then the strip pseudo-distance between the families of curves

f(t) := D0(x(·, t), y(·, t)), t ∈ U,

is a non-increasing function of t ∈ U .

Proof. By the triangle inequality of the strip pseudo-distance it follows

|f(t1)− f(t2)| ≤ D0(x(·, t1), x(·, t2)) +D0(y(·, t1), y(·, t2)).

Therefore, by definition 5.1.b) f : U → R is Holder 1/2 and Lipschitz con-tinuous away from zero. In particular, there exists a unique continuous ex-tension f : [0, T ] → R. Since the strip pseudo-distance is invariant under re-parametrization, by definition 5.1.b) and theorem 4.33 the upper right Diniderivative of f satisfies

D+t f(t) ≤ 0,

for every t ∈ U . Since f is Lipschitz in [t0, T ], t0 > 0 by Rademacher it is almosteverywhere differentiable and there holds

f(t1)− f(t2) =

∫ t2

t1

D+t f(t) dt,

for all 0 < t1 ≤ t2 ≤ T . The lemma follows, since U is a full measure set of[0, T ] and f is continuous at t = 0.

Definition 5.10. Let x0 ∈ C0(S1,Rn). A weak solution (x, U) on S1 × [0, T ]satisfies the initial condition x0, if

limt→0t∈U

x(·, t) = x0,

uniformly on S1.

Theorem 5.11. Let (x, Ux), (y, Uy) be weak solutions on S1 × [0, T ] satisfyingthe initial condition x0, y0 respectively. If D0(x0, y0) = 0, then (x, Ux) and(y, Uy) are equivalent in the sense of definition 5.3.

126

Proof. Let U := Ux ∩ Uy be the common full measure set in [0, T ]. By thetriangle inequality of the strip pseudo-distance and corollary 4.20 there holds

limt→0t∈U

D0(x(·, t), y(·, t)) = D0(x0, y0) = 0.

Therefore, by theorem 5.9 the strip pseudo-distance between x(·, t) and y(·, t)vanishes for all t ∈ U . By definition of a weak solution, there exist smoothimmersed curve shortening flows x, y : S1 × U → Rn such that for every con-nected component V of U there exist non-decreasing, degree one Lipschitz mapsφV , φV : S1 → S1 such that

x(u, t) = x(φV (u), t), (u, t) ∈ S1 × V,

y(u, t) = y(φV (u), t), (u, t) ∈ S1 × V.

In particular, x(·, t) and y(·, t) are smooth curves with bounded curvature forall t ∈ U . By lemma 4.43 these curves do not contain hairs for all t ∈ U .Hence by lemma 4.27 there exists a homeomorphism ψt : S1 → S1 such thatx(ψt(u), t) = y(u, t). Since both satisfy the curve shortening equation thishomeomorphism is constant on every connected component of U . Hence onevery connected component V of U there holds

x(u, t) = x(φV (u), t), (u, t) ∈ S1 × V,y(u, t) = x(ψV (u), t), (u, t) ∈ S1 × V,

where ψV = ψt φV for some t ∈ V .

5.3 A Convergence Theorem

The following lemma has some analogy to the proof of ramp convergence fromAltschuler and Grayson [8].

Lemma 5.12. Let xi : S1 × [0, Ti) → Rn be a sequence of maximal smoothcurve shortening flows such that the initial curves xi(·, 0) converges in C1 to acurve x0 ∈ C1(S1,Rn).

There for every t0 ≥ 0 and δ > 0 with t0 + δ < T := limTi, there existt1 ∈ [t0, t0 + δ] and i0 ∈ N such that the curvature ki(·, t) of xi(·, t) is boundedby

supS1

|ki(·, t)|2 ≤C

ε2

for all i ≥ i0 and t ∈ [t1 + ε2/2, t1 + ε2], where ε := 10−40δ/2L(x0) and C is anabsolute constant.

Proof. Since the family at time t = 0 converges in C1, there exists i0 such thatfor all i ≥ i0 there holds |xiu(·, 0)| ≤ 2 supS1 |(x0)u|. Moreover, by lemma 3.3 itfollows

|xiu(·, t)| ≤ 2 supS1

|(x0)u|,

L(xi(·, t)) ≤ C1 := 2L(x0),

for all i ≥ i0 and 0 ≤ t < Ti.

127

Let t0, δ > 0 such that t0+δ < T . We may assume that there holds t0+δ < Tifor all i ≥ i0. Moreover, by lemma 4.19, we can choose i0 big enough such thatthe strip pseudo-distance between the initial curves satisfies

D0(xi(·, 0), xj(·, 0)) ≤ c, (154)

for all i, j ≥ i0, where c = c(C1, C1, η, ε, 10−20) > 0 is given by theorem 4.46with η := 10−16ε2/C1. Note that by theorem 4.33 this estimate is preservedunder the flow. For the smooth flow xi(·, t) there holds∫ Ti

0

∫S1

|ki(·, t)|2dµitdt ≤ L(xi(·, 0)) ≤ C1.

Therefore, there exists a time t1 ∈ [t0, t0 + δ] such that∫S1

|ki0(·, t1)|2dµi0t1 ≤C1

δ.

By corollary 3.38 and the choice ε = 10−40δ/C1, the curve xi0(·, t1) is a Lipschitzε−graph and there holds

Lipε(xi0(·, t1)) = tan

(√C1ε

4δ

)≤ 10−20.

Hence, by lemma 3.45 there holds Lipε(xi0(·, t)) ≤ 10−8 for all times in t ∈

[t1, t1 + ε2].Now we consider xi(u, t) for i ≥ i0. Analogously, we find a time t2 ∈

[t1, t1 + ε2/4] such that ∫S1

|ki|2dµit2 ≤4C1

ε2,

By corollary 3.38 it follows Lipη(xi(·, t2)) ≤ 10−8, where η = 10−16ε2/C1.The estimate (154) together with theorem 4.46 implies that the curve xi(·, t2)

is in fact a Lipschitz ε−graph, and we have the estimate

Lipε(xi(·, t2)) ≤ 20Lipε(x

i0(·, t2)) + 12Lipη(xi(·, t2) ≤ 10−6.

Hence, by lemma 3.46 the curvature is bounded by

supS1

|ki(·, t)|2 ≤C

t− t2,

for all t ∈ [t2, t2 + ε2]. This implies the following uniform curvature bound onthe time interval J := [t2 + ε2/4, t2 + ε2]

supS1×J

|ki|2 ≤C

ε2. (155)

Such an estimate holds for all i ≥ i0, but the interval J = J(i) depends on i.Since t2 ∈ [t1, t1 + ε2/4] there holds the inclusion

I := [t1 + ε2/2, t1 + ε2] ⊂ [t2 + ε2/4, t2 + ε2].

Therefore, the uniform curvature bound (155) holds for all curves xi(·, t) withi ≥ i0 in the common time interval I.

128

Corollary 5.13. Let xi : S1 × [0, Ti) → Rn be a sequence of maximal smoothcurve shortening flows such that the initial curves xi(·, 0) converges in C1 to acurve x0 ∈ C1(S1,Rn). Then the open set

U :=

t ∈ (0, T ) : ∃ε > 0, sup

i≥1sup

S1×(t−ε,t+ε)|ki| <∞

is dense in [0, T ], where T := limTi.

Proof. We argue by contradiction and assume there exists an open interval(t′0, t

′1) ⊂ [0, T ] \U . By lemma 5.12 with δ := min1, L(x0) min1, (t′1− t′0)/2

and t0 = t′0, we have a contradiction, since δ+ε2 ≤ δ+(δ/L(x0))2 ≤ t′1− t′0.

Definition 5.14. A sequence xi : S1× [0, Ti)→ Rn of smooth curve shorteningflows Z−converge to a pair (x, U) consisting of an open dense set U of [0, T ]and a family of Lipschitz maps x(·, t) : S1 → Rn for t ∈ U , if T = limTi,

xi → x in C0loc(S

1 × U),

and for every connected component V ⊂ U there exist orientation preservingdiffeomorphisms φiV : S1 → S1, a non-decreasing, degree one Lipschitz mapφV : S1 → S1 and smooth curve shortening flows xiV , xV : S1 × I → Rn suchthat

φiV → φV in C0(S1),

xiV → xV in C∞loc(S1 × V ),

and there holds xi(u, t) = xiV (φiV (u), t) for all (u, t) ∈ S1 × V .Moreover, we say that (x, U) is a Z−limit of smooth solutions to the initial

condition x0 ∈ C1(S1,Rn), if additionally there holds xi(·, 0)→ x0 in C1(S1).

Theorem 5.15. Let xi : S1 × [0, Ti) → Rn be a sequence of smooth curveshortening flows such that the initial curves xi(·, 0) converges in C1 to a curvex0 ∈ C1(S1,Rn). Then there exist a map x : S1 × U → Rn, where U is theopen dense set of [0, T ] given in corollary 5.13, and a subsequence xij whichZ−converges to (x, U) in the sense of definition 5.14.

Moreover, (x, U) satisfies the conditions 5.1.a), b) and c) of a weak solution.

Proof. By taking a subsequence we can assume that Ti converges to T := limTi.Let B ⊂ U be a countable dense set of [0, T ]. For fixed t0 ∈ B we parametrizethe sequence xi(·, t0) by constant speed. There exist orientation preservingdiffeomorphisms φit0 : S1 → S1 and smooth maps xit0 : S1 → Rn such that

xi(u, t0) = xit0(φit0(u)), u ∈ S1,

|Dvxit0(v)| = ci(t0), u ∈ S1,

For i big enough there hold Lip(xi(·, 0)) ≤ 2Lip(x0), T − t0 ≤ 2(Ti − t0) andL(xi(·, 0)) ≤ 2L(x0). By lemma 3.3 we have L(xi(·, t0)) ≤ L(xi(·, 0)) ≤ 2L(x0)and Lip(xi(·, t0)) ≤ Lip(xi(·, 0)) ≤ 2Lip(x0) Since every curve with length L iscontained in a ball with radius L, theorem 3.11 implies√

T − t0 ≤√

2(Ti − t0) ≤ L(xi(·, t0)).

129

Since Lip(xi(·, t0)) = ci(t0)Lip(φit0) and 2πci(t0) = L(xi(·, t0)), for i bigenough there hold √

T − t02π

≤ ci(t0) ≤ L(x0)

π, (156)

and

Lip(φit0) ≤4πLip(x0)√T − t0

.

By Arzela-Ascoli there exist a subsequence ij , a Lipschitz map φt0 : S1 → S1

and a Lipschitz map xt0 : S1 → Rn such that

xijt0 → xt0, in C0(S1), (157)

φijt0 → φt0, in C0(S1). (158)

By a diagonal sequence we may assume that this hold for every t0 in the count-able set B. Note that since φit0 are bijective and orientation preserving maps,

φt0 is a non-decreasing, degree one map of S1.We now claim that xij converges locally uniformly to a continuous limit on

S1 × U . Since from now we will not take any further subsequence, we drop thej for simplicity.

Let V be a connected component of U . Since B is dense in [0, T ] and V isopen, there holds B ∩ V 6= ∅. For a chosen t0 ∈ B ∩ V we define φiV := φit0and

xiV (v, t) := xi((φiV )−1(v), t), (u, t) ∈ S1 × V.

We fix a compact interval I = [t1, t2] ⊂ V , such that t0 ∈ I and t1 ∈ B.Note that every compact interval of V is contained in an interval of this form.

Since I ⊂ U , by definition of the set U , there exist ε > 0 and C0 such that

supi≥1

supS1×[t1−ε,t2+ε]

|ki(u, t)| ≤ C0.

Together with theorem 3.6 this implies that for every l ≥ 1 there exists aconstant Cl depending only on l, C0 and ε such that

supi≥1

supS1×I

∣∣Dlski∣∣ ≤ Cl.

By lemma 3.7 and (156), for every (m, l) ∈ N2 \0 there exists a constant Km;l

depending only on (m, l), T − t0 and Ci for i = 0, ..., 2m+ l − 1 such that

supi≥1

supS1×I

∣∣Dmt D

lux

iV (v, t)

∣∣ ≤ ci(t0)−lKm;l ≤(2π)lKm;l

(T − t0)l/2.

Hence the restriction of the maps xiV to S1 × I sub-converges to a smoothmap xI : S1 × I → Rn. By lemma (3.3) and (156) for all (v, t) ∈ S1 × I thereholds

|DvxiV (v, t)| ≥ e−C

20 (t2−t0)ci(t0) ≤ e−C

20T√T − t0/(2π).

In the limit this implies

|DvxI(v, t)| ≥ e−C20T√T − t0/(2π), (v, t) ∈ S1 × I.

130

Therefore, xI(·, t) : S1 → Rn is an immersion for all t ∈ I, and x satisfies thecurve shortening equation on S1 × I. Since t1 ∈ B, the map xI is the uniquecurve shortening flow with the smooth initial condition xt1. Hence the limitdoes not depend on the subsequence. This shows that the full limit

limi→∞

xiV |S1 × I = xI ∈ C∞(S1 × I),

exists. Since this holds for every compact interval I ⊂ V with t0 ∈ I and t1 ∈ B,this shows

xiV → xV , in C∞loc(S1 × V ), (159)

where xV (v, t) := xI(v, t), for (v, t) ∈ S1 × I.Since this holds for every connected component V ⊂ U , this defines the map

x : S1 × U → Rn by x(v, t) := xV (v, t), (v, t) ∈ S1 × V . Note that x dependson each connected component V on the choice t0 ∈ V and the limit φt0.

We define the map x : S1 × U → Rn by

x(u, t) := x(φV (u), t), (u, t) ∈ S1 × V.

The map x is well defined, since by (158) and (159) it is the unique limit

xi(u, t) = xiV (φiV (u), t)→ x(u, t), (u, t) ∈ S1 × V.

This shows that a subsequence xi converges in C0loc(S

1 × U) to x. Moreover, itproves that xi Z−converges to (x, U) in the sense of definition 5.14.

The properties 5.1.a), b) and c) are proven in the following lemmata.

Lemma 5.16. Let (x, U) be a Z−limit of smooth solutions as in definition 5.14.Then the Radon measures µit converges to µt for all t ∈ U and the pair (x, U)satisfies condition 5.1.a) of a weak solution, i.e. the Radon measures µt satisfiesµt1 ≥ µt2 for all t1, t2 ∈ U with t1 ≤ t2.

Proof. Let H ⊂ S1 be a measurable set. By the smooth convergence of xi(·, t)to x(·, t) and the area formula it follows

µit(H) :=

∫H

|xiu(u, t)|du =

∫φi(H)

|xiu(u, t)|du

→∫φ(H)

|xu(u, t)|du (i→∞)

=

∫H

|xu(u, t)|du =: µt(H).

By lemma 3.3 there holds µit1 ≥ µit2 for all 0 ≤ t1 ≤ t2 ≤ Ti. Hence themonotonicity of µt in U follows by the convergence.

Lemma 5.17. Let (x, U) be a Z−limit of smooth solutions to the initial condi-tion x0 ∈ C1(S1,Rn) as in definition 5.14. Then (x, U) satisfies condition 5.1.b)of a weak solution, i.e. the family x(·, t), t ∈ U is Holder 1/2 and Lipschitz con-tinuous away from the start time with respect to the strip pseudo-distance.

131

Proof. For the smooth approximation xi there holds∫ t0

0

∫S1

|ki|2dµiτdτ ≤ L(xi(·, 0))

for every t0 > 0 there exists τ0 ∈ [0, t0] such that∫S1

|ki|2 dµiτ0 ≤L(xi(·, 0))

t0.

By the monotonicity of the total absolute curvature and the length it follows∫S1

|ki|dµit0 ≤∫S1

|ki|dµiτ0

≤(∫

S1

|ki|2 dµiτ0

)1/2

L(xi(·, τ0))1/2

≤ L(xi(·, 0))

t1/20

.

Claim. x(·, t), t ∈ U is Holder 1/2 continuous at the start time with respectto the strip pseudo-distance.Proof. For 0 < t1 ≤ T there holds

D0(xi(·, t1), xi(·, 0)) ≤∫ t1

0

∫S1

|ki|dµitdt

≤∫ t1

0

L(xi(·, 0))

t1/2dt

≤ 2L(xi(·, 0))t1/21 .

By corollary 4.20, and uniform convergence xi(·, t) → x(·, t), t ∈ U , the strippseudo-distance converges. Together with the C1 convergence of the initialcurve it follows

D0(x(·, t1), x(·, 0)) ≤ 2L(x0)t1/21 ,

for all t1 ∈ U .Claim. x(·, t) is Lipschitz continuous away from the start time with respect

to the strip pseudo-distance.Proof. For 0 < t0 ≤ t1 ≤ t2 ≤ T there holds

D0(xi(·, t1), xi(·, t2)) ≤∫ t2

t1

∫S1

|ki| dµitdt

≤∫S1

|ki| dµit0 |t1 − t2|

≤ L(xi(·, 0))

t1/20

|t1 − t2|.

Analogously, for t1, t2 ∈ U with 0 < t0 ≤ t1 ≤ t2 ≤ T it follows

D0(x(·, t1), x(·, t2)) ≤ L(x(·, 0))

t1/20

|t1 − t2|.

132

Lemma 5.18. Let (x, U) be a Z−limit of smooth solutions as in definition 5.14.If U is the open dense set of [0, T ] given in corollary 5.13, then (x, U) satisfiesthe 5.1.c) of a weak solution.

Proof. By assumption for every connected component V of U there exist asmooth curve shortening flow xV : S1 × V → Rn and a non-decreasing, degreeone Lipschitz map φV : S1 → S1 such that

x(u, t) = xV (φV (u), t), (u, t) ∈ S1 × V.

Hence we only have to show that the smooth flow xV can not be continuedbeyond t∗ = supV , if t∗ < T . Otherwise, there exist M < ∞ and t0 < t∗ suchthat

supS1×[t0,t∗)

|k(xV )| ≤M.

the curvature of xV is uniformly bounded in S1 × [t0, t∗). By the smooth con-vergence xiV (·, t)→ xV (·, t) for t ∈ V , for every t1 ∈ [t0, t∗) there holds

limi→∞

supS1

|ki(·, t1)| ≤M.

By theorem 3.6, there hold

limi→∞

supS1×[t1,t1+1/(8M)]

|ki| ≤ 2M.

Note that since t∗ < T we may assume that t∗ + 1/(8M) < T . Hence fort1 > t∗ − 1/(8M) this implies t∗ ∈ U which contradicts the assumption that Vis a connected component of U .

5.4 Extension to Singular Times and Regularity

Our aim is to extend a the map x : S1 × U → Rn to a map on S1 × [0, T ].

Definition 5.19. Let U be an open dense set of (0, T ) and x(·, t) : S1 → Rn,t ∈ U be a family of Lipschitz curves. For t0 ∈ (0, T ], we define the left limits,if they exist by

x+(u, t0) := limt→t0

t∈U∩[0,t0]

x(u, t), µ+t0 := lim

t→t0t∈U∩[0,t0]

µt,

and for t0 ∈ [0, T ) the right limits, if they exist by

x−(u, t0) := limt→t0

t∈U∩[t0,T ]

x(u, t), µ−t0 := limt→t0

t∈U∩[t0,T ]

µt.

For convenience, we use the convention x+(·, 0) := x−(·, 0) and µ+0 := µ−0 .

By definition for all t ∈ U there hold x+(u, t) = x−(u, t) = x(u, t) andµ+t = µ−t = µt. By the lower semi-continuity of the length functional there holdsL(x+(·, t0)|I) ≤ µ+

t0(I) and L(x−(·, t0)|I) ≤ µ−t0(I) for every interval I ⊂ S1.

The notation is maybe misleading since x+(·, t0) is the limit t → t−0 , but thenext lemma justifies this convention.

133

Definition 5.20. Let µt be family of Radon measures on S1 for t ∈ U , whereU is a dense subset of [0, T ]. Assume the family is non-increasing in U , i.e.µt1 ≤ µt2 for t1 ≤ t2. We define the mass drop measure ν on S1 × (0, T ) by

ν(I × (t1, t2)) := µt1(I)− µt2(I),

where t1, t2 ∈ U , t1 < t2 and I ⊂ S1 is an open interval. By Carathodory’sextension theorem, this induces an outer measure via

ν(E) := inf

∑i≥1

ν(Ai) : E ⊂⋃i≥1

Ai, Ai ∈ A

, E ⊂ S1 × (0, T ),

where A := I × (t1, t2) : t1, t2 ∈ U, I ⊂ S1 open interval.

By lemma 5.16, the mass drop measure ν is defined for every Z−limit ofsmooth solutions. In this case µt is absolutely continuous with respect toLebesgue measure L1 on S1 for all t ∈ U . Hence the σ−algebra of ν mea-surable sets contains the σ−algebra induced by the product of the Borel sets on(0, T ) with L1 measurable set on S1.

Lemma 5.21. Let (x, U) be a Z−limit of smooth solutions to the initial condi-tion x0 ∈ C1(S1,Rn) as in definition 5.14. Then the limits µ+

t0 , x+(·, t0) exist

for every t0 ∈ (0, T ], and the limits µ−t0 , x−(·, t0) exist for every t0 ∈ [0, T ).Moreover, for all 0 ≤ t1 < t2 ≤ T , 0 < s1 ≤ s2 < T and L1 measurable setI ⊂ S1 there hold

µ−t1(I)− µ+t2(I) = ν(I × (t1, t2)), (160)

µ+s1(I)− µ−s2(I) = ν(I × [s1, s2]). (161)

Furthermore, for all 0 ≤ t1 < t2 ≤ T with |t1 − t2| ≤ µ+t2(S1)3/µ0(S1) and

u ∈ S1 there holds∣∣x−(u, t1)− x+(u, t2)∣∣ ≤ 2ν(S1 × (t1, t2)) + 3µ0(S1)1/3|t1 − t2|1/3, (162)

and for all 0 < s1 ≤ s2 < T with |s1 − s2| ≤ µ−s2(S1)3/µ0(S1) and u ∈ S1 thereholds∣∣x+(u, s1)− x−(u, s2)

∣∣ ≤ 2ν(S1 × [s1, s2]) + 3µ0(S1)1/3|s1 − s2|1/3. (163)

Proof. Since U is dense, for every t0 ∈ (0, T ] there exists a sequence ti ∈ U withti t0. By the compactness of Radon measures, we may assume there exists aRadon measure µ∗ such that

µ∗ = limi→∞

µti .

By the monotonicity µt1 ≥ µt2 for all t1, t2 ∈ U with t1 ≤ t2, the full limit exists

µ+t0 = lim

t→t0t∈U∩[0,t0]

µt = inft∈U∩[0,t0]

µt,

and since µ+t0 = µ∗, this limit is a Radon measure. By the C1 convergence at

time t = 0, there holds µ0(S1) ≤ L(x0). By a similar argument as above thereexists the full limit

µ−t0 = limt→t0

t∈U∩[t0,T ]

µt = supt∈U∩[t0,T ]

µt,

134

for every t0 ∈ [0, T ). The inequalities (160) and (161) follow immediately bythe definition of ν and monotone convergence

ν(I × (t1, t2)) = limt′1t1t′1∈U

limt′2t2t′2∈U

ν(I × (t′1, t′2)) = lim

t′1t1t′1∈U

µt′1(I)− limt′2t2t′2∈U

µt′2(I),

ν(I × [s1, s2]) = lims′1s1s′1∈U

lims′2s2s′2∈U

ν(I × (s′1, s′2)) = lim

s′1s1s′1∈U

µs′1(I)− lims′2s2s′2∈U

µs′2(I).

For I ⊂ S1 and t1, t2 ∈ U , 0 < t1 < t2 < T there holds∫ t2

t1

∫I

|ki|2 dµitdt = µit1(I)− µit2(I)→ ν(I × (t1, t2)), (i→∞).

Hence the Radon measures dνi := |ki(u, t)|2dµitdt on S1 × [0, Ti] converges tothe mass drop measure ν on S1 × (0, T ).

We claim that the limit x+(u, t0) exists and argue similarly as in the proofof lemma 3.21. We apply corollary 3.20 to the smooth approximation to obtain∣∣xi(u, t1)− xi(u, t2)

∣∣ ≤ 2

∫ t2

t1

∫S1

|ki|2 dµitdt+ 3µi0(S1)1/3|t1 − t2|1/3,

for every u ∈ S1, and 0 ≤ t1 ≤ t2 < Ti. In the limit this implies the inequality

|x(u, t1)− x(u, t2)| ≤ 2ν(S1 × (t1, t2)) + 3µ0(S1)1/3|t1 − t2|1/3, (164)

for every u ∈ S1, and t1, t2 ∈ U with 0 ≤ t1 ≤ t2 < T .Let tj ∈ U be a sequence such that tj t0. Since ν(S1 × (tj , tj+1))→ 0 as

(j →∞), by inequality (164) x(u, tj) is a Cauchy sequence. Since this holds forevery sequence tj , the full limit x+(u, t0) exists for all u ∈ S1. Since the familyx(·, t) is uniformly Lipschitz for t ∈ [0, T ], this limit is a Lipschitz map. Theexistence of the limit x−(u, t0) follows analogously.

Finally, inequality (162) follows in the limit t′1 ∈ U t1 and t′2 ∈ U t2by the estimate (164). Analogously, inequality (163) follows in the limit s′1 ∈U s1 and s′2 ∈ U s2.

Corollary 5.22. Let (x, U) be a Z−limit of smooth solutions to the initial con-dition x0 ∈ C1(S1,Rn) as in definition 5.14. If the length µ+

t (S1) is continuousat t = t0, then there holds x+(·, t0) = x−(u, t0). Moreover, there exists a count-able subset B of [0, T ] such that the maps x+, x− are continuous in S1×[0, T ]\B.

Proof. The map t 7→ x+(·, t) is by definition continuous from the left. Hence wehave to prove that for all u ∈ S1 there holds,

|x+(u, t0)− x+(u, t1)| → 0, (t1 t0).

We fix u ∈ S1 and t0 ∈ [0, T ). Note that µ+t0(S1) > 0, since otherwise t0 = T .

Since µ+t (S1) is continuous at t0, by the equalities (160) and (161) there

holdsν(S1 × [t0, t1)) = µ+

t0(S1)− µ+t1(S1)→ 0, (t1 t0).

Hence by inequality (163) there holds∣∣x+(u, t0)− x−(u, t0)∣∣ ≤ 2ν(S1 × t0) = 0.

135

By definition of x+ and x−, this implies that x+, x− are continuous in S1×t0.By the equalities (160) and (161) t 7→ µ+

t (S1) is a monotonically non-increasingfunction. Hence µ+

t (S1) is continuous in a co-countable set.

Our next goal is to prove that a Z−limit (x, U) of smooth solutions satisfiesthe last condition of a weak solution, i.e. U is a full measure set of [0, T ].

In lemma 3.51 we have seen that if the curvature of a smooth flow is boundedin a past neighborhood of (u0, T∗) ∈ S1, where T∗ denotes the first singular time,then u0 ∈ R∗ is a regular point. Next we prove that (u0, t0) is a past-regularpoint of a Z−limit, if the curvature of a sequence of smooth flow is uniformlybound in a past neighborhood of (u0, t0).

Definition 5.23. For (u0, t0) ∈ S1 × [0, T ) and ρ > 0 we define the interval

I+t0(u0, ρ) :=

⋃[a, b] ⊂ S1 : u0 ∈ [a, b], µ+

t0([a, u0]) = ρ = µ+t0([u0, b])

.

As the union of closed connected sets this defines a closed interval. Sinceµ+t0 is absolutely continuous with respect to the Lebesgue measure on S1, there

holds µ+t0(v) = 0 for every point v ∈ S1.

Therefore, there holds µ+t0([a, u0]) = ρ = µ+

t0([u0, b]) for I+t0(u0, ρ) = [a, b].

Lemma 5.24. Let (x, U) be a Z−limit of smooth solutions xi : S1 × [0, Ti) →Rn. Let u0 ∈ S1 and t0 ∈ (0, T ], where T := limTi, and assume there existconstants ρ > 0, δ ∈ (0, t0) and C0 <∞, such that

sup(u,t)∈I×[t0−δ,ti∗)

|ki(u, t)| ≤ C0, (165)

where I := I+t0(u0, ρ) and ti∗ := mint0, Ti.

Then there exist a non-decreasing Lipschitz map φ : J → R and a smoothflow x : V × [t0 − δ/2, t0]→ Rn, where V := φ(J), such that there holds

x(φ(u), t) = x+(u, t), (u, t) ∈ J × [t0 − δ/2, t0],

where J := I+t0(u0, ρ/2). In particular, there holds (u0, t0) ∈ Rp.

Proof. By the curvature bound (165) and lemma 3.51 the smooth flows xi ex-tends on the interval I = [a, b] smoothly to ti∗. Hence we may assume that (165)holds on I × [t0− δ, ti∗]. By theorem 3.40 for every l ≥ 1 and ρ0 > 0 there existsa constant Cl depending only on l, C0 and ρ0 such that

|Dlski| ≤ Cl, (u, t) ∈ Ωi,

where Ωi := (u, t) ∈ I × [t0 − δ + δ0, ti∗] : µit([a, u]), µit([u, b]) ≥ ρ0 and

δ0 := ρ0/(10C0) and I = [a, b].

We choose ρ0 := mine−C20δρ/2, 5C0δ and claim that there exists i0 such

that there holdsJ × [t0 − δ/2, ti∗] ⊂ Ωi, i ≥ i0.

For the time interval we observe δ0 ≤ δ/2. Now we estimate the distancebetween J = [a′, b′] and I = [a, b]. Fix t ∈ [t0 − δ/2, ti∗]. By lemma 3.3, for

136

t′ ∈ U ∩ [t0 − δ/2, ti∗] it follows

µit([a, a′]) ≥ e−C

20δ/2µit′([a, a

′])→ e−C20δ/2µt′([a, a

′]), (i→∞)

→ e−C20δ/2µ+

t0([a, a′]), (t′ ∈ U t0)

= e−C20δ/2

(µ+t0([a, u0])− µ+

t0([a′, u0]))

= e−C20δ/2ρ/2 > ρ0.

Therefore, there exists i0 such that µit([a, a′]) ≥ ρ0 holds for all i ≥ i0 and

t ∈ [t0 − δ/2, ti∗]. Analogously, we get µit([b, b′]) ≥ ρ0 for all i ≥ i0 and t ∈

[t0 − δ/2, ti∗]. This proves J × t ⊂ Ωi for all i ≥ i0 and t ∈ [t0 − δ/2, t0].We parametrize by arc-length at t = t0 − δ/2. Since xi(·, t0 − δ/2) is an

immersion, the map φi : J → R defined by

φi(u) :=

∫ u

u0

|xiu(v, t0 − δ/2)|dv,

is a diffeomorphism onto is image Vi := φi(J), and the map

xi(v, t) := xi(φ−1i (v), t), (v, t) ∈ Vi × [t0 − δ/2, ti∗],

defines a smooth curve shorting flow with |xiv(v, δ/2)| = 1.By lemma 3.7 for every (m, l) ∈ N2 \ 0 there exists a constant Km,l de-

pending only on (m, l), T∗ and Ci for i = 0, ..., 2m+ l − 1 such that

|Dmt D

lvxi(v, t)| ≤ Km,l, (v, t) ∈ Vi × [t0 − δ/2, ti∗].

Since Ci only depends on i, C0 and ρ0, and ρ0 only depends on C0, δ and ρ, theconstant Km,l in fact only depends on (m, l), C0, δ and ρ.

Since 0 ≤ Duφi = |xiu| ≤ Lip(x0) by Arzela-Ascoli φi sub-converges to a non-decreasing Lipschitz map φ : J → V := φ(J) ⊂ R. The maps xi sub-convergesto a smooth map x : V × [t0 − δ/2, t0] → Rn. By lemma (3.3) there holds

|xv(v, t)| ≥ e−C20δ/2 for all (v, t) ∈ V × [t0− δ/2, t0]. Therefore, x(·, t) : V → Rn

is an immersion for all t ∈ [t0 − δ/2, t0], and x satisfies the curve shorteningequation on V × [t0 − δ/2, t0]. Since by assumption xi → x in C0

loc(S1 × U),

there holds

x(φ(u), t) = x(u, t), (u, t) ∈ J × [t0 − δ/2, t0] ∩ U.

Since the left hand side is a continuous map on J × [t0 − δ/2, t0], this implies

x(φ(u), t) = x+(u, t), (u, t) ∈ J × [t0 − δ/2, t0].

Definition 5.25. We define the space-time map X(u, t) := (x+(u, t), t) ∈ Rn×R, the space-time track Γ = X(S1 × [0, T ]), and denote by Σp := X(Sp) the setof past-singular space-time points.

Although, there is no reason to prefer the left limit x+ : S1 × [0, T ] → Rn,we choose it as the unique extension of (x, U). However, that the space-timetrack Γ is a closed set, it is essential to define it with x+. We shortly argue for

137

this fact. Let (yi, ti) ∈ Γ, i ≥ 1 be a sequence that converges to (y0, t0). Bya subsequence, we may assume that either ti ≤ t0 or ti ≥ t0. In the first case,(y0, t0) ∈ Γ follows by definition of x+. In the second case, by lemma 3.14 itfollows x−(S1, t) ⊂ x+(S1, t). Together with the definition of x− this implies(y0, t0) ∈ x−(S1, t0) ⊂ Γ. Therefore, the space-time track Γ is a closed set.

The measure on space-time Rn × R, which is natural to the problem, is the1−dimensional Hausdorff measure with respect to the parabolic metric

dP ((x1, t1), (x2, t2)) := max|x1 − x2|, |t1 − t2|1/2.

We denote this measure by H1;1/2, since it can be written as the product of1−dimensional Hausdorff measure on Rn with the 1/2−dimensional Hausdorffmeasure on R, where both are with respect to the Euclidean metric. Moreprecisely, for Ω ⊂ Rn × R we define

H1;1/2(Ω) := supδ>0H1;1/2δ (Ω),

where diamP (U) := supdP (p, q) : p, q ∈ U and

H1;1/2δ (Ω) := inf

∑i≥1

diamP (Ui) : Ω ⊂⋃i≥1

Ui, diamP (Ui) ≤ δ

.

Theorem 5.26. Let (x, U) be a Z−limit of smooth solutions to the initial con-dition x0 ∈ C1(S1,Rn) as in definition 5.14. Then the set of the past-singularspace-time points Σp has parabolic Hausdorff dimension less or equal than 1.

Moreover, for every D0 there exists a constant C, depending only on D0,such that if the maximal area ratio of x0 satisfies Θ(x0) ≤ D0, then the parabolicHausdorff measure of Σp is bounded by

H1;1/2(Σp) ≤ CL(x0).

Proof. Assume (x, U) is the Z−limit of smooth flows xi : S1× [0, Ti)→ Rn. Wedenote the maximal area ratio of the approximation by

Θi(t) := supp∈Rn

supr>0

µit((xi)−1(Br(p), t))

2r.

Since xi(·, 0) → x0 in C1(S1), there holds Θi(0) → Θ(x0). Hence there existsi0 such that for all i ≥ i0 there holds Θi(0) ≤ Θ(x0) + 1 ≤ D0 + 1.

Let ε0 = ε0(D0 + 1) > 0 be as in the local regularity theorem 3.61. Wedefine the set

Ω :=

(y, t) ∈ Γ : lim

r→0limi→∞

r−1

∫(Xi)−1(Pr(y,t))

|ki|2 dµiτdτ < ε20, t > 0

,

where Xi(u, t) := (xi(u, t), t), (u, t) ∈ S1 × [0, Ti) is the space-time map of xi

and Pr(y, t) := Br(y)× (t− r2, t) is an open parabolic cylinder.Claim 1. The set Ω is contained in the set of past-regular space-time points, i.e.

Ω ⊂ Γ \ Σp.

138

Proof. For fixed (y0, t0) ∈ Ω there exist r0 ∈ (0,√t0) and a subsequence ij such

that there holds

r−10

∫(Xij )−1(Pr0 (y0,t0))

|kij |2 dµijτ dτ ≤ ε20,

for all j ≥ 1. We drop the j for simplicity. By the theorem 3.61, there existsC0, depending only on D0, such that

|ki(u, t)| ≤ C0/r0, (166)

holds for all i ≥ 1 and (u, t) with Xi(u, t) ∈ Pr0/2(y0, t0).Fix u0 ∈ S1 such that X(u0, t0) = (y0, t0), i.e. x+(u0, t0) = y0.Claim 2. There exists i0 such that there holds xi(u, t) ∈ Br0/2(y0) for

all i ≥ i0 and (u, t) ∈ I × [t0 − δ, t0], where δ := (r0/2)2 min1, 1/C0, and

I := I+t0(u0, ρ) as in definition 5.23 with ρ := e−(C0/r0)2δ(r0/5).

We first show that claim 2 implies claim 1. By claim 2 and lemma 5.24 thereholds (u0, t0) ∈ Rp. Since this holds for all u0 ∈ S1 with X(u0, t0) = (y0, t0), itfollows (y0, t0) ∈ Γ \Σp. Since (y0, t0) in Ω was arbitrary this shows Ω ⊂ Γ \Σpand proves claim 1.

Proof of claim 2. Fix ε > 0 and let I(1) := [a, u0], I(2) := [u0, b], whereI = [a, b]. By definition of x+ and µ+, there exists t1 ∈ [t0 − δ, t0] such that

µt1(I(j))− µ+t0(I(j)) ≤ ε and |x(u, t1)− x+(u, t0)| ≤ ε.

Since t1 ∈ U , there exists i0 such that for all i ≥ i0 there holds

|µt1(I(j))− µit1(I(j))| ≤ ε and |xi(u, t1)− x(u, t1)| ≤ ε.

To prove xi(u, t) ∈ Br0/2(y0) for all (u, t) ∈ I × [t0 − δ, t0], we show δ ≤ δ for

δ := sups > 0 : xi(u, t) ∈ Br0/2(y0), i ≥ i0, t ∈ [t0 − s, t0],

u ∈ I = I+t0(u0, %), % = e−(C0/r0)2s(r0/5)

.

We assume by contradiction that δ < δ and fix (u, t) ∈ I × [t0 − δ, t0]. By thetwo estimates on the right hand side above, there holds

|xi(u, t)− y0| ≤ |xi(u, t)− xi(u0, t)|+ |xi(u0, t)− xi(u0, t1)|+ |xi(u0, t1)− x(u0, t1)|+ |x(u0, t1)− x+(u0, t0)|≤ µit(I(j)) + |xi(u0, t)− xi(u0, t1)|+ 2ε. (167)

By (166) there holds |ki| ≤ C0/r0 on I × [t0 − δ, t0] for all i ≥ i0. By this

curvature bound, lemma 3.3 and since δ < δ, for the first term of (167) itfollows

µit(I(j)) ≤ e(C0/r)

2δµit1(I(j)) ≤ e(C0/r)2δ(µt1(I(j)) + ε)

≤ e(C0/r)2δ(µ+

t0(I(j)) + 2ε) ≤ e(C0/r)2δ(ρ+ 2ε)

≤ r0/5 + 2eC0/4ε.

139

The second term of (167) is bounded by

|xi(u0, t)− xi(u0, t1)| ≤ (C0/r0)δ ≤ r0/4.

For ε small enough this implies |xi(u, t) − y0| ≤ (19/20)r0/2. Since this holds

for all (u, t) ∈ I × [t0 − δ, t0] and i ≥ i0, this contradicts the maximality of δand proves claim 2.

Note that Γ0 := Γ ∩ t = 0 consists by definition only of past-singularpoints, but since H1;1/2(Γ0) = 0, this does not affect not the statement of thetheorem. By claim 1, the set of past-singular space-time points for positivetimes Σp ∩ t > 0 is contained in the set

A :=

(y, t) ∈ Γ : lim

r→0limi→∞

r−1

∫(Xi)−1(Pr(y,t))

|ki|2dµiτdτ ≥ ε20, t > 0

.

Fix δ > 0. For every point (y0, t0) ∈ A there exists r0 = r0(y0, t0) ∈ (0, δ)such that

limi→∞

r−10

∫(Xi)−1(Pr0 (y0,t0))

|ki|2dµiτdτ ≥ ε20/2.

The set A is covered by the parabolic balls

BPr0(y0, t0) := (y, t) ∈ Rn × R : dP ((y, t), (y0, t0)) < r0, (y0, t0) ∈ A,

where r0 = r0(y0, t0) ∈ (0, δ) is as above.By Vitali’s covering lemma for Rn × R with the parabolic metric dP , there

exist rj ∈ (0, δ), (yj , tj) ∈ A for j ≥ 1 such that the parabolic balls BPrj (yj , tj)are pairwise disjoint and there holds

A ⊂⋃j≥1

BP5rj (yj , tj),

limi→∞

r−1j

∫(Xi)−1(Prj (yj ,tj))

|ki|2dµiτdτ ≥ ε20/2.

Since the parabolic cylinders Prj (yj , tj) ⊂ BPrj (yj , tj) are also pairwise disjoint

and diamP (BP5rj (yj , tj)) = 10rj ≤ 10δ, it follows

H1,1/210δ (Σp ∩ t > 0) ≤ H1,1/2

10δ (A) ≤∑j≥1

10rj

≤ 20ε−20 lim

m→∞

m∑j=1

limi→∞


|ki|2dµiτdτ

≤ 20ε−20 lim

m→∞limi→∞

m∑j=1


|ki|2dµiτdτ

≤ 20ε−20 lim

i→∞

∫ Ti

0

∫S1

|ki|2 dµiτdτ

≤ 20ε−20 lim

i→∞L(xi(·, 0)) = 20ε−2

0 L(x0).

Together with H1,1/210δ (Σp ∩ t = 0) ≤ 10δL(x0), in the limit δ → 0 this implies

H1;1/2(Σp) ≤ CL(x0),

where C only depends on ε0, and ε0 only depends on D0.

140

Corollary 5.27. Let (x, U) be a Z−limit of smooth solutions to the initialcondition x0 ∈ C1(S1,Rn) as in definition 5.14. Then the open dense set U of[0, T ] satisfies H1/2([0, T ] \ U) ≤ CL(x0).

In particular, U is a L1 full measure set of [0, T ].

Proof. Since (x, U) satisfies condition 5.1.c) of a weak solution, by lemma 5.8 wehave U = R = Rp ∩ [0, T ). The time projection of the past-singular space-timepoints are the past-singular times, i.e. Sp = π(Σp). Hence by theorem 5.26 itfollows

H1/2([0, T ] \ U) = H1/2(Sp) = H1/2(π(Σp)) ≤ H1,1/2(Σp) ≤ CL(x0).

In [11, 12] K. Deckelnick has proven a similar result. He defines weak so-lutions for the curve shortening flow in arbitrary co-dimension, by the equa-tion xt = xuu/|xu|2 in a classical L2 sense. This equation differs only by aparametrization from the curve shortening flow. He shows that the set of sin-gular times, for solution which occurs as limit of smooth flows, has Hausdorff-dimension less or equal than 1/2. Deckelnick’s weak solutions are not knownto be unique and this regularity only holds for the limit of smooth flows. Theadvantage of our weak solution is that by lemma 5.5 all equivalent solutions hasthe same set of regular times.

5.5 Existence

Definition 5.28. For an immersed initial curve x0 ∈ C1(S1,Rn) and a sequenceλi 0, we define the initial curves

xi0(u) :=(x0(u), λi cos(u), λi sin(u)

)∈ Rn+2.

The maximal smooth curve shortening flows

xi : S1 × [0, Ti)→ Rn+2,

with initial condition xi(·, 0) = xi0, is called a wave approximation.

This approximation is an alternative to the approximations with ramps in-troduced by Altschuler and Grayson [8]. For a comparison, we review the rampapproximation in appendix A. The main advantage of ramps is that their flowexists forever. This ensures that the limit of ramp flows handles all singulari-ties at once. An advantage of the wave approximation is that they are closedcurves, and we can use the strip pseudo-distance directly for them. A furtheradvantage of the wave approximation is that the density ratio in Rn+2 is uni-formly bounded. This allows us to apply the regularity theorem 3.61 directly.Clearly, the wave approximation does not exist forever, but since the projectionπR2xλ(·, 0) is an embedded strictly convex curve theorem 3.59 implies that theflow exists until it converges to a point or a hair. This ensure that the limitflow overcomes all singularities up to the extinction time.

Lemma 5.29. Let (x, U) be a Z−limit of a wave approximation. Then theimage of x(·, t) : S1 → Rn+2 lies in Rn × 0 for all t ∈ U , and T ∗ := supU isthe extinction time in the sense of definition 5.2, i.e. there holds

limt→T∗t∈U

D0(x(·, t)) = 0.

141

Proof. By theorem 3.11 the smooth flow of xi(·, t) stays between the verticalhyperplanes E(ν,−σ) and E(ν, σ), for all normal vectors ν ∈ 0n × R2 andσ > λi. Hence there holds

xi(S1, t) ⊂ Y ∈ Rn × R2 : |πR2(Y )| ≤ λi,

and the limit curve lies in the plane Rn × 0.Let tj ∈ U be a sequence with tj T ∗. Since limTi = T ∗, there exists i0(j)

such that for all i ≥ i0(j) there holds tj < Ti. By the triangle inequality of thestrip pseudo-distance, for all i ≥ i0(j) there holds

D0(x(·, tj)) ≤ D0(xi(·, Ti)) +D0(xi(·, Ti), xi(·, tj)) +D0(xi(·, tj), x(·, tj)).

Since the projection of the wave approximation πR2xi(u, 0) = λi(cos(u), sin(u))is an immersed uniformly convex curve, by theorem 3.59 the limit

xi(·, Ti) = limtTi

xi(·, t),

is contained in a line. Hence the first term on the right hand side vanishes.Together with lemma 4.19 and lemma 4.31 this implies

D0(x(·, tj)) ≤ C|Ti − tj |+ C‖xi(·, tj)− x(·, tj)‖C0 ,

where the constant C only depends on x0. In the limit (i→∞) this implies

D0(x(·, tj)) ≤ C|T ∗ − tj | → 0, (j →∞).

Theorem 5.30. For every immersed curve x0 ∈ C1(S1,Rn) there exists amaximal weak solution (x, U) with initial condition x0.

Proof. Let xi : S1× [0, Ti)→ Rn+2 be a wave approximation to the initial curvex0, as in definition 5.28. Then xi(·, 0) converges in C1(S1) to the immersedinitial curve (x0, 0, 0). Since x0 is an immersed C1 curve, there exists ε > 0such that Lipε(x0) ≤ 10−7. Hence the short time existence of a weak solutionfollows by theorem 3.47 and in particular there holds T ∗ = limTi > 0.

By theorem 5.15 a subsequence xij converges to a pair (x, U), which satisfiesconditions 5.1.a), b) and c) of a weak solution. By corollary 5.27 the open denseset U is a full measure of [0, T ∗] and hence (x, U) is a weak solution.

By lemma 5.29, this solution is maximal in the sense of definition 5.2 andthe image x(S1, t) lies in Rn.

By lemma 5.21 the limit x−(·, 0) exists. By corollary 3.20 for i0 big enoughand i ≥ i0 there holds∣∣xi(·, t)− xi0∣∣C0 ≤ 2

∫ t

0

∫S1

|ki|2 dµiτdτ + 3µi0(S1)1/3t1/3

≤ 4ν(S1 × (0, t)) + 6L(x0)1/3t1/3 → 0, (t→ 0).

This allows us to estimate

|x−(·, 0)− x0|C0 ≤ |x−(·, 0)− x(·, t)|C0 + |x(·, t)− xi(·, t)|C0

+ |xi(·, t)− xi0|C0 + |xi0 − x0|C0 .

We first choose t ∈ U small enough such that first and third term are small forall i ≥ i0 and afterwards choose i ≥ i0 big enough the second and fourth termis small. Therefore, (x, U) satisfies the initial condition x−(·, 0) = x0.

142

5.6 Further Regularity

Related to the definition 5.23, we define maximal intervals in the domain S1.

Definition 5.31. For the Radon measures µ+t and µ−t and a point u ∈ S1 we

define the maximal intervals

I+t (u) :=

⋃I ⊂ S1 : I is connected, u ∈ I, µ+

t (I) = 0,

I−t (u) :=⋃I ⊂ S1 : I is connected, u ∈ I, µ−t (I) = 0.

Since there holds µ±t∗(v) = 0 for every point v ∈ S1, the maximal intervalsare closed. Note the it possible that the maximal interval consists only of thepoint u and therefore we use the convention that a point is an interval.

Definition 5.32. By Rp(t), Sp(t) we denote the past-regular and past-singularpoints in S1 at time t. i.e. there hold u ∈ Rp(t) if and only if (u, t) ∈ Rp andu ∈ Sp(t) if and only if (u, t) ∈ Sp.

We denote the space slice of the past-singular space-time points at time t,by Σp(t) := x+(Sp(t), t). Note that this set can also be written by

Σp(t) = x ∈ Rn : (x, t) ∈ Σp,

where Σp is the set of past-singular space-time points, see definition 5.25.

Lemma 5.33. Let (x, U) be a weak solution with initial condition x0 ∈ C1(S1,Rn).Then the total absolute curvature

K(t) :=

∫S1

|k| dµt,

is a monotonically non-increasing function of t ∈ U .

Proof. For t ∈ U the curve x(u, t) = x(φ(u), t) is, up to a non-decreasing, degreeone map, smooth and total absolute curvature is well define. Moreover, the mapK : U → R only depends on the equivalence class of the weak solution. Henceby existence and uniqueness of weak solutions, we may assume that (x, U) is aZ−limit of smooth solutions. By lemma 3.4 the total absolute curvature of thesmooth approximation are non-increasing. Since xi(·, t) smoothly convergenceto x(·, t) for t ∈ U , K(t) is a non-increasing function of t ∈ U .

By lemma 5.33 the left limit of the total absolute curvature

K+(t0) := limt→t0

t∈U∩[0,t0]

K(t),

exists for all t0 ∈ (0, T ].

Theorem 5.34. Let (x, U) be a Z−limit of smooth solutions to the initial con-dition x0 ∈ C1(S1,Rn) as in definition 5.14. Then for each time t∗ ∈ (0, T ]the set of past-singular points Sp(t∗) of (x, U) consists of finitely many disjointclosed intervals

Sp(t∗) =

N⋃i=1

Ii,

143

and there exists a constant c > 0 depending on nothing such that

N ≤ K+(t∗)

c.

Moreover, the map x+(·, t∗) is constant on each Ii and hence Σp(t∗) is the unionthe N points pi = x+(Ii, t∗).

We expect that the theorem actually holds for c = π, as in theorem 3.52 forthe first singular time. This would follow if we could improve theorem 3.30 toarbitrary Lipschitz constant. See Angenent [6] for the co-dimension one case.

Proof. We fix c > 0 such that tan(c/2) = 1/100. For t ∈ U we define the Radonmeasure κt on S1 by

κt(ψ) :=

∫S1

ψ(u)|k(u, t)|dµt(u),

for ψ ∈ C0(S1).By the compactness of Radon measures there exists a non-decreasing se-

quence tj ∈ U , tj t∗ such that the limit

κ∗ = limj→∞

κtj ,

exists. By lemma 3.4 the total absolute curvature is non-increasing and hencethe limit measure is bounded by

κ∗(S1) = lim

j→∞κtj (S

1) = K+(t∗).

We define the collection of maximal intervals

B := I+t∗(u) : κ∗(I

+t∗(u)) ≥ c, u ∈ S1

and the set A :=⋃B. Since two maximal intervals are by definition either equal

or disjoint, the number N of maximal interval in B is bounded by

cN ≤ κ∗(A) ≤ K+(t∗).

We denote these maximal intervals by I1, ..., IN . Hence the set A is the unionof N disjoint closed intervals Ii and in particular is a closed set. We define theopen set W := S1 \A.

Claim 1. There holds W ⊂ Rp(t∗).Proof. We fix u∗ ∈W and write I := [u1, u2] := I+

t∗(u∗) for the maximal interval.By definition of W , there holds κ∗(I) < c. Since the total measure of κ∗ is finite,for the decreasing sequence of closed intervals Jk := [u1 − 1/k, u2 + 1/k] thereholds

limk→∞

κ∗(Jk) = κ∗(I).

Hence there exists a closed neighborhood J = [v1, v2] of I such that κ∗(J) < c.By definition of I both connected components H1, H2 of J \ I has positive µ+

t∗measure and we define

ε := mini=1,2

µ+t∗(Hi).

144

By the convergence of the Radon measures κtj we have

limj→∞

κtj (J) ≤ κ∗(J) < c.

There exists j0 ≥ 1 such that for σ := ε/1000 there holds

κtj0 (J) < c, (168)

2ν(S1 × (tj0 , t∗)) ≤ σ/5, (169)

3µ0(S1)1/3(t∗ − tj0)1/3 ≤ σ/5. (170)

We assume that (x, U) is the Z−limit of smooth flows xi : S1× [0, Ti)→ Rnand xi(·, 0) → x0 in C1. By this convergence, inequalities (168), (169), (170)and since tj0 ∈ U , there exists i0 such that for all i ≥ i0 there holds∫

J

|ki|dµitj0 < c, (171)

2

∫ t∗

tj0

∫S1

|ki|2dµitdt ≤ σ/4, (172)

3µi0(S1)1/3|t∗ − tj0 |1/3 ≤ σ/4, (173)

|xi(·, tj0)− x(·, tj0)|C0 ≤ σ/10. (174)

We define δ := t∗ − tj0 .Claim 2. For all u ∈ S1, t ∈ [t∗ − δ, t∗] and i ≥ i0 there holds

|xi(u, t)− x+(u, t∗)| ≤ σ, (175)

Proof. Fix u ∈ S1, t ∈ [t∗ − δ, t∗] and i ≥ i0. By lemma 5.21, (169) and (170)there holds

|x(u, tj0)− x+(u, t∗)| ≤ 2ν(S1 × (tj0 , t∗)) + 3µ0(S1)1/3(t∗ − tj0)1/3 ≤ 2σ/5.

By corollary 3.20, (172) and (173) there holds

|xi(u, t)− xi(u, tj0)| ≤ 2

∫ t

tj0

∫S1

|ki|2dµitdt+ 3µi0(S1)1/3|t− tj0 |1/3 ≤ σ/2.

Together with (174) these estimates imply the claim 2 via

|xi(u, t)− x+(u, t∗)| ≤ |xi(u, t)− xi(u, tj0)|+ |xi(u, tj0)− x(u, tj0)|+ |x(u, tj0)− x+(u, t∗)|≤ σ/2 + σ/10 + 2σ/5 = σ.

By lemma 3.37, inequality (171) and the choice of c > 0, for each i ≥ i0 thesegment xi(J, tj0) is the graph of a Lipschitz function ri : Vi ⊂ Li → L⊥i , with

Lip(ri) ≤ η1 := tan(c/2) = 1/100,

where Vi = [ai, bi] is an interval on a line Li.By (175) and lemma 3.39, for each i ≥ i0 and t ∈ [tj0 , t∗] the segment

xi(Ji(t), t) is the graph of a Lipschitz function ri(·, t) : Vi ⊂ Li → L⊥i with

145

Lip(ri(·, t)) ≤ η2 := 10η1 = 1/10, where Vi := [ai, bi] := [ai + ∆, bi − ∆],∆ := σ/(2η1) and Ji(t) := u ∈ J : πLi(x

i(u, t)) ∈ Vi.Theorem 3.30 implies the uniform curvature bound

|ki(u, t)|2 ≤2

δ+C

ε2, (176)

for all i ≥ i0, t ∈ [t∗ − δ/2, t∗] and u ∈ Ji(t) := u ∈ J : πLi(xi(u, t)) ∈ Vi,

where Vi := [ai, bi] := [ai + ε/5, bi − ε/5].We define the interval J := [v1, v2] by µ+

t∗([v1, u∗]) = ε/2 = µ+t∗([u∗, v2]).

Claim 3. There holds J ⊂ Ji(t) for all t ∈ [t∗ − δ, t∗] and i ≥ i0.Proof. Fix i ≥ i0 and t ∈ [t∗−δ, t∗]. We have to show that πLi(x

i(J , t)) ⊂ Vi.Fix u ∈ J . By (175) there holds

πLi(xi(u, t)) ≤ πLi(xi(u∗, tj0)) + |xi(u∗, tj0)− x+(u∗, t∗)|

+ |x+(u∗, t∗)− x+(u, t∗)|+ |x+(u, t∗)− x(u, t)|≤ πLi(xi(u∗, tj0)) + |x+(u∗, t∗)− x+(u, t∗)|+ 2σ.

By lower semi-continuity of the length functional and since u ∈ J there holds

|x+(u∗, t∗)− x+(u, t∗)| ≤ L(x+(·, t∗)|[u∗, u]) ≤ µ+t∗([u∗, u]) ≤ ε/2.

Since xi(J , tj0) is the graph of ri, there holds

πLi(xi(v2, tj0))− πLi(xi(u∗, tj0)) ≥

µtj0 ([u∗, v2])√1 + η2

1

≥µ+t∗([u2, v2])√

1 + η21

≥ ε√1, 0001

.

Since bi = bi −∆− ε/5, ∆ = 50σ and bi = πLi(xi(v2, tj0)), we arrive at

πLi(xi(u, t)) ≤ bi −

ε√1, 0001

+ ε/2 + 2σ

= bi + 50σ + ε/5− ε√1, 0001

+ ε/2 + 2σ

≤ bi + 52σ − ε/5.

Since σ = ε/1000, this shows πLi(xi(u, t)) ≤ bi. The inequality ai ≤ πLi(xi(u, t))

follows analogously. This proves claim 3.By 176 and claim 3, for all i ≥ i0 there holds

|ki(u, t)| ≤ C, (u, t) ∈ J × [t∗ − δ/2, t∗].

Hence lemma 5.24 implies (u∗, t∗) ∈ Rp. By definition there holds u∗ ∈ Rp(t∗)and hence W ⊂ Rp(t∗). This proves claim 1.

Now we claim the opposite inclusion Rp(t∗) ⊂ W . We fix u∗ ∈ Rp(t∗).By definition there exist δ > 0, a neighborhood H of u∗, a non-decreasing,surjective Lipschitz map φ : H → (0, 1), and a smooth curve shortening flowx : (0, 1)× (t∗ − δ, t∗]→ Rn, such that

x+(u, t) = x(φ(u), t), (u, t) ∈ H × (t∗ − δ, t∗].

146

By definition for I := I+t∗(u∗) there holds µ+

t∗(I) = 0 and since x(·, t∗) is asmooth immersion, the map φ is constant on I. Therefore, there holds

κ∗(I) = limj→∞

∫I

|k(x(u, tj))|dµtj (u) = limj→∞

∫φ(I)

|k(x(u, tj))|dµtj (u) = 0,

which implies u∗ ∈W .Therefore, there holds Rp(t∗) = W and

Sp(t∗) =

N⋃i=1

Ii,

where Ii ⊂ S1 are closed intervals with µ+t∗(Ii) = 0.

The past-singular space points Σp(t∗) := x+(Sp(t∗), t∗) is the union of thepoints pi := x+(Ii, t∗), 1 ≤ i ≤ N .

147

Appendices

A Ramps

We follow the work of Altschuler and Grayson [8], where they use ramps todefine a global solution to the curve shortening flow in the plain.

Definition A.1. A curve X : R→ Rn+1 is called ramp, if its tangent vector Thas positive vertical component at all points i.e. 〈T, V 〉 > 0. A ramp is calledperiodic, with period 2π, if there exists λ > 0 such that X(u+2π) = X(u)+λV .

For initial data x0 : S1 → Rn and λ > 0, we define the periodic ramp

Xλ0 (u, 0) := (x0(eiu), λu) ∈ Rn+1, u ∈ R.

Let Xλ : S×[0, Tλ)→ Rn+1 be the smooth curve shortening flow, which satisfies

d

dtXλ(u, t) = k(Xλ)(u, t),

with the initial condition Xλ(·, 0) = Xλ0 . Since the ramps are periodic the

projection of this solution to Rn, defines a family of closed curves

xλ(u, t) = πRn(Xλ(u, t)), u ∈ [0, 2π].

Remark. These projections are solutions of the regularized equation

xλt =xλuu

|xλu|2 + λ2−⟨xλuu, x

λu

⟩xλu

(|xλu|2 + λ2)2,

with the initial condition xλ(·, 0) = x0. It is also possible to use the LeraySchauder principle and standard elliptic estimates, to show that a smooth solu-tion to this regularized problem exists.

A major feature of ramps is the flow exists for all positive times.

Theorem A.2. (Altschuler-Grayson [8]) Let X : S1 × [0, T∗) → Rn+1 be asmooth curve shortening flow, and assume that the initial curve X(·, 0) is aperiodic ramp. Then

1) X(·, t) is a ramp for all t ≥ 0,

2) the curvature |k| is bounded for all time, and hence T∗ =∞,

3) X(·, t) converges to a straight line in infinite time.

Proof. Since X(·, 0) is a ramp its tangent vector T satisfies 〈T, V 〉 > 0 for someunit vector V . The evolution equation of this quantity is

∂

∂t〈T, V 〉 =

∂2

∂s2〈T, V 〉+ k2 〈T, V 〉 .

By the maximum principle the minimum is a non-decreasing function of time,

t 7→ minu∈[0,2π]

〈T (u, t), V 〉 .

149

Hence 〈T, V 〉 stays positive and X(·, t) is a ramp with respect to the samevertical vector V , for all t ≥ 0. Moreover, the ratio |k|/ 〈T, V 〉 is well define.The evolution equation of this ratio is

∂

∂t

|k|〈T, V 〉

=

(∂2

∂s2

|k|〈T, V 〉

)+

2

〈T, V 〉

(∂

∂s〈T, V 〉

)(∂

∂s

|k|〈T, V 〉

)− τ2

1

|k|〈T, V 〉

,

where τ1 is the first torsion of X(·, t). Therefore, the maximum of the ratio is anon-increasing function of time and we have the curvature estimate

|k(u, t)| ≤ |k(u, t)|〈T (u, t), V 〉

≤ maxu∈[0,2π]

|k(u, 0)|〈T (u, 0), V 〉

.

Thus the flow exists for all time. By integration over one period of the rampthere holds ∫ ∞

0

∫ 2π

0

|k|2dµtdt ≤ L (X([0, 2π], 0)) ,

and since ddt

∫ 2π

0|k|2 dµt is bounded, we get

limt→∞

∫ 2π

0

|k|2 dµt = 0.

Hence the curvature k tends to zero as t → ∞. Therefore, X(·, t) converges toa straight line as t→∞.

This is a impressive long time existence result with an amazing short proof.The problem in using a ramps to approximate closed curves is, that the strippseudo-distance is not defined for ramps since they are not closed curves. Toprove that the limit of a ramp approximation unique one has to measure thedistance between two ramps. For an area estimate between two periodic ramps,one has to connect these curves by two short arcs. The resulting curve boundsa disk type surface, but not the whole boundary satisfies the curve shorteningflow. Usually the disk-size decreases by D+

t D0(x(·, t)) ≤ −2π. The prize we payto hold the connecting arcs at the periodic position, is that the disk-size onlyis non-increasing. Altschuler and Grayson overcome this difficulty by taking alarge number of turns of the ramps. The projection of the resulting disk-typesurface to Rn is a multi-covered annulus and its area tends to zero as the numberof turns increases. In higher co-dimension this projection is a surface in Rn, andit turns out that it is difficult to get an accurate estimate of it.

150

B Minimal Surfaces

Since the most results about minimal surfaces uses concept of Jordan curves,we shortly induce them. The original definition of a Jordan curve is a curve inR2 which is homeomorphic to S1. Jordan proved that such curves divides theplane into two components, an inside and an outside region. This is called theJordan theorem and seems to be obvious, but Jordan curves does not have tobe rectifiable and include fractal curves which are nowhere differentiable. Theconcept of Jordan curves is also used for curves in Rn, although there exists noanalog to Jordan theorem for n ≥ 3.

Definition B.1. A Jordan curve γ in Rn is a curve which is homeomorphic toS1 .

Remark. Sometimes Jordan curve are also called simple closed curve, or closedJordan curves. In the later case, Jordan curve in Rn, is define as a curve whichis homeomorphic to a interval [a, b]. We will call such not closed Jordan curves,Jordan arcs.

Since the interval S1 is compact, each continuous injective map x : S1 → Rnis a homeomorphism onto the image. Hence, a Jordan curve could be define asthe image of a continuous injective map x : S1 → Rn.

We may define a Jordan curve as a equivalence class of continuous injectivemap x : S1 → Rn, modulo diffeomorphism of S1, and oriented Jordan curvesby mode out only orientation preserving diffeomorphism.

A Jordan curve γ is called rectifiable ifH1(γ) <∞. Note that a Jordan curveis rectifiable if and only if there exists a Lipschitz continuous representativex : S1 → Rn.

The general Plateau Problem in Rn can be formulated as follows. Givendisjoint, oriented Jordan curves γk in Rn, k = 1, ..., N , and a compact, oriented,two dimensional manifold M , with N boundary components. Does there existsa map F : M → Rn such that ∂F parametrizes the curves γk, such that Area(F )is minimal under this constraints.

We may think of a minimal surface F : M → Rn as a critical point of thearea functional. By the first variation formula this are surfaces with vanishingmean curvature H. Each surface F admits a conform representation, and forconform maps there holds ∆F = HFu ∧ Fv. Hence, a conform map F is acritical point of the area functional if and only if F is harmonic.

The Dirichlet energy E of a map F ∈ H1,2(M,Rn) is define by

E(F ) =1

2

∫M

|Fu|2 + |Fv|2dudv.

Generally, there holds Area(F ) ≤ E(F ), and for conform maps F there holdsequality. Hence, if F minimizes the Dirichlet energy, then F is conform andminimizes the area.

This motivates the following definition.

Definition B.2. Let γk, k = 1, ..., N be disjoint, oriented Jordan curves in Rn.A map F ∈ H1,2(M,Rn) ∩ C0(M,Rn) is a solution to the Plateau problem forγk, if M is a compact, oriented, two dimensional manifold, with N boundarycomponents ∂M = σ1 ∪ ... ∪ σN , and

F is harmonic in, i.e. ∆F = 0 in M (177)

151

F is conform, i.e.|Fu|2 = |Fv|2, 〈Fu, Fv〉 = 0 in M (178)

F |σk : σk → γk is a weakly monotone oriented parametrization. (179)

For γ := (γk)k=1,..,N let S(γ) be the set of maps F ∈ H1,2(M,Rn) ∩C0(M,Rn), such that M is compact, oriented, two dimensional manifold, andF satisfies the condition (179) holds.

For g ≥ 0 we define

e(γ, g) := infE(F ) : F ∈ S(γ, g),

where S(γ, g) consists of all maps F ∈ S(γ) such that M is connected and hasgenus g,

e∗(γ, g) := infE(F ) : F ∈ S∗(γ, g),

where S∗(γ, g) consists of all maps F ∈ S(γ) such that M is either connectedand has genus less than g, or is disconnected and has genus g.

The following result is from Douglas [15]. Good references are the booksfrom Dierkes, Hildebrandt, Kuster and Wohlrab [14] and Nitsche [47].

Theorem B.3. Let γk, k = 1, ..., N be disjoint, rectifiable, oriented Jordancurves in Rn. If e(γ, g) < e∗(γ, g) then there exists a solution F ∈ S(γ, g) of thePlateau problem according to definition B.2 . Moreover, the solution satisfiesE(F ) = e(γ, g).

We will use this theorem of two curves γ1, γ2 and genus g = 0. There alwaysexists two disk type minimal surfaces bounded by γ1 resp. γ2, which minimizesthe area in this class. If the infimum of the area over annulus type surfaces isstrictly smaller than of two disk type surfaces, then there exist an annulus typeminimal surface, which minimizes the area in this class.

Minimal surfaces satisfies the following monotonicity for the area ratio. LetF : Mm ⊂ Rn be a properly embedded minimal surface. The density quotientat x0 ∈ Rn with radius r > 0 is define by

Θ(x0, r) :=Hm(M ∩Br(x0))

ωmrm,

where ωm denotes the volume of the unit ball in Rm. Since we assume that Fis proper, by Sard’s theorem and the co-area formula it follows that the mapr 7→ Θ(x0, r) is absolute continuous. Hence the derivative exists for almostevery r > 0 and by calculate this one can prove the following lemma.

Lemma B.4. (Monotonicity of the area ratio) If ∂M ∩ Bρ(x0) = ∅, then forevery 0 < r < R < ρ there holds

Θ(x0, R) = Θ(x0, r) +1

ωm

∫M∩BR(x0)\Br(x0)

|(D|x− x0|)⊥|2

|x− x0|mdHm(x).

152

C Injective Approximation

We follow the first capture of the book from Guillemin and Pollack [30]. In thissection M denotes a manifold maybe with boundary ∂M , and P a manifoldwithout boundary, and Q a submanifold of P .

Definition C.1. A smooth map f : M → P is said to be transversal to Q atthe point x ∈ f−1(Q), if there hold

dfx(TxM) + Tf(x)Q = Tf(x)P. (180)

The map f is transversal to the submanifold Q, if this holds for every pointx ∈ f−1(Q). In this case we use the notation f t Q.

Let g : P → Rd be a submersion such that Q = g−1(0) and f t Q. Then ify = f(x) ∈ Q and we have

Rd = dgy(TyP ) = dgy(dfx(TxM) + TyQ) = dgy(dfx(TxM)).

Therefore, g f is a submersion at a point x if and only if f is transversal to Qat x. Hence, if f is transversal to Q, then f−1(Q) is a submanifold of M . If Mhas boundary and the restriction ∂f = f |∂M is also transversal to Q i.e.

dfx(Tx(∂M)) + Tf(x)Q = Tf(x)P,

for all x ∈ f−1(Q) ∩ ∂M , the same argument implies that f−1(Q) ∩ ∂M is asubmanifold. Moreover, in this case the pre-image f−1(Q) is a manifold withboundary

∂(f−1(Q)) = f−1(Q) ∩ ∂M.

Now, we vary the map f by a smooth function F : M × S → Rn withF (x, s) = fs(x), where the parameter set S is a manifold without boundary.

Theorem C.2. Let M be a manifold with boundary ∂M , and P be a manifoldwithout boundary. Suppose F : M × S → P is a smooth map, and Q is asubmanifold of P with ∂Q ∩ F (M × S) = ∅. If both F and ∂F are transversalto Q, then for almost every s ∈ S, both fs and ∂fs are transversal to Q.

Proof. Since F and ∂F are transversal to Q, the pre-image W = F−1(Q) is asubmanifold with boundary ∂W = f−1(Q)∩∂M . The restriction of the naturalprojection π : M ×S → S to the submanifolds W and ∂W defines smooth mapsπ : W → S and ∂π : ∂W → S. By Sard’s theorem almost every s ∈ S is aregular value of both maps π and ∂π.

The goal is to show that if s ∈ S is a regular value of π, then fs is transversalto Q. Let y = fs(x) ∈ Q. We will show that if s ∈ S is a regular value of π,then there holds

dF(x,s)(T(x,s)(M × S)) ⊂ d(fs)x(TxM) + TyQ.

Since F t Q we have dF(x,s)(T(x,s)(M × S)) + Ty(Q) = TyP . Together thisimplies the transversality condition d(fs)x(TxM) + Ty(Q) = Ty(P ).

Let (w, e) ∈ T(x,s)(M × S) = TxM + TsS. Since dπ(x,s) : T(x,s)W → TsS issurjective, there exists (u, e) ∈ T(x,s)W such that dπ(x,s)(u, e) = e. Therefore,we have

dF(x,s)(w, e) = dF(x,s)(w − u, 0) + dF(x,s)(u, e) = d(fs)x(w − u) + dF(x,s)(u, e),

153

and since F : W → Q, there holds dF(x,s)(u, e) ∈ TyQ. This shows the desiredinclusion.

The same argument shows that if s ∈ S is a regular value of ∂π, then ∂fs istransversal to Q.

Corollary C.3. If M,N are submanifolds of Rn without boundaries, then foralmost every v ∈ Rn there holds (M + v) t N .

Proof. In the case P = Rn, we can choose S to be the open unit ball in Rn.For any given map f : M → Rn, the map F : M × S → Rn, F (x, s) = f(x) + sis a submersion and hence transversal to any submanifold Q. Therefore, withM = f(x) and Q = N the theorem implies the corollary.

The corollary can be improved to general maps f : M → P , where P issubmanifold of Rn without boundary. In this case can use the closest pointprojection π : P ε → P , which is define on an open neighborhood P ε = x ∈Rn : ∃y ∈ P, |x−y| < ε(y). Hence, we have the submersion F (x, s) = π(f(x)+ε(f(x))s), where again S is the open unit ball in Rn.

We improve the corollary in an other way.

Theorem C.4. Let M be a manifold with boundary ∂M , and let C be closedsubset of M . Let f : M → Rn be a smooth map, and let Q be a submanifold ofP with δ = distRn(∂Q, f(M)) > 0. Suppose f t Q on C. Then there exists asmooth cutoff function φ : M → [0, 1] with supp(φ) ⊂M \ C such that the mapg(x) = f(x) + φ(x)v and ∂g are transversal to Q for almost every v ∈ Rn with|v| < δ.

Proof. First we show that there exists an open neighborhood U of C such thatf t Q on U . Let x ∈ C and x ∈ f−1(Q), since f is continuous M \ f−1(Q)is an open neighborhood of x, on which f t Q trivially holds. If x ∈ C andx ∈ f−1(Q) we take a submersion g : P → Rd such that Q = g−1(0). Sincef t Q at x, the map g f is regular at x, and therefore also in a neighborhoodof x. Hence f is transversal to Q on this neighborhood of x. Now, we take asmooth cutoff function τ : M → [0, 1] such that τ = 0 on C, and τ = 1 on U c.We define φ = τ2 and F (x, v) = f(x) + φ(x)v for v ∈ Rn with |v| < δ. Weclaim that F is transversal to Q. Let (x, v) ∈M ×Rn such that F (x, v) ∈ Q. Ifφ(x) 6= 0, then F is a submersion at (x, v), and hence transversal to Q at thispoint. If φ(x) = 0 by construction dφx = 2τ(x)dτx = 0. Therefore, there holdsdF(x,v)(u,w) = dfx(u), where (u,w) ∈ TxM × Rn. By construction φ(x) = 0implies x ∈ U , and therefore we have f t Q at x. This shows that F istransversal to Q at (x, v), since

dF(x,v)(TxM + Rn) + TF (x,v)Q = dfx(TxM) + Tf(x)Q = Rn.

This shows that F t Q, and an analog argument shows that also ∂F t Q holds.By assumption we have F (M × Bnδ (0)) ∩ ∂M = ∅. Therefore, by theorem C.2both F (., v) and ∂F (., v) are transversal to Q, for almost every v ∈ Bnδ (0).

Now, we analyze self intersections of one map.

Definition C.5. Let M,P be a manifolds without boundaries. Two mapsf, g : M → P has a transversal intersection point at f(x) = g(y) if

dfx(TxM) + dgy(TyM) = Tf(x)P.

154

If this holds for every intersection point we call f transversal to g and use thenotation f t g. Analog a self intersection point x 6= y with f(x) = f(y) is calledtransversal if

dfx(TxM) + dfy(TyM) = Tf(x)P.

Theorem C.6. Let M be manifold without boundary, and let f : M → Rn bean immersion. Assume that there exists finitely many open precompact subsetsVk ⊂ M , 1 ≤ k ≤ N such that their closures are disjoint i.e. V k ∩ V l = ∅ fork 6= l, and such that for all x, y ∈ M with x 6= y and f(x) = f(y) there existk 6= l with x ∈ Vk and y ∈ Vl. Then there exist ε > 0 and smooth cutoff functionsφk : M → [0, 1] with supp(φk) ⊂ Vk such that for every v1 ∈ Bnε (0) there exists aset of Ln measure zero N2 = N2(v1) ⊂ Rn, and for every v2 ∈ Bnε (0) \N2 thereexists a set of Ln measure zero N3 = N3(v1, v2), and so on there exists a set ofLn measure zero NN = NN (v1, ..., vN−1) such that for every vN ∈ Bnε (0) \NNthe map

f(x) +

N∑k=1

φk(x)vk,

only has transversal self intersection points.

Proof. Let B := x ∈ M : ∃y 6= x, f(x) = f(y) be the set where f is notinjective. By assumption there holds B ⊂ ∪Nk=1Vk. Since B is a closed set, byUrysohn’s lemma we find open subsets Uk such that B ∩Vk ⊂ Uk and Uk ⊂ Vk.

By assumption the restrictions of f to Vk are injective maps. Therefore, theimage f(V k) is an embedded submanifold with boundary. As in theorem C.4 foreach 1 ≤ k ≤ N , there exists a smooth cutoff function φk with supp(φk) ⊂ Ukand φk = 1 on B ∩ Uk, and such that φk = 0 implies d(φk)x = 0. We definethe map gk(v) : V k → Rn by gk(v, x) = f(x) + φkv, where v ∈ Rn with |v| < ε.Since f embeds the compact set V k in Rn, we have the positive constant

α = inf

|f(x)− f(y)|distM (x, y)

: x, y ∈ V k, x 6= y, 1 ≤ k ≤ N> 0.

Provided that |v| < ε ≤ α/2 the maps gk(v) are injective on V k. Moreover,since f is an immersion we have

β = inf

|dfx||d(φk)x|

: x ∈ V k, 1 ≤ k ≤ N> 0.

If we additionally choose ε < β/2 the maps gk(v) are immersions, and henceembeddings.

We define the closed set C = M \ ∪kVk and the positive constant

γ = mindistRn(f(C), f(Uk) : 1 ≤ k ≤ N.

We additionally choose ε ≤ γ/2, and therefore have gk(vk, Uk) ∩ f(C) = ∅.Let g(x) = f(x) +

∑Nk=1 φk(x)vk and assume that

|vk| < ε ≤ minα/2, β/2, γ/2.

We claim that if g(x) = g(y) for x 6= y, then there exists k 6= l such that x ∈ Vkand y ∈ Vl. Since g = f on M \ ∪Uk is injective, we can assume that x ∈ Uk.

155

By the assumption ε ≤ γ/2 we have y /∈ C, and since g is injective on Vk thereholds y /∈ Vk.

Since f has no self intersection points in the compact sets V k \ Uk, thecontinuous images has positive distance

δ = mindistRn(f(V k \ Uk), f(V l \ Ul) : k 6= l > 0.

For every k 6= l provided that |vk|, |vl| < ε ≤ δ/3 this implies

distRn(gk(v, V k \ Uk), gl(v, V l \ Ul)) > δ/3.

Hence for fixed k ∈ 1, ..., N, provided that |vl| < ε = minα/2, β/2, γ/2, δ/3for l 6= k, by theorem C.4 the map gk(vk) is transversal to gl(vl, V l) for almostevery |vk| < ε.

We fix v1 ∈ Bnε (0). By the last statement there exists set of Ln measure zeroN2 = N2(v1) such that for every v2 ∈ Bnε (0) \N2 the map g2(v2) is transversalto g1(v1, V 1). We iterate this argument until we find set of Ln measure zeroNN = NN (v1, ..., vN−1) such that for every vN ∈ Bnε (0) \NN the map gN (vN )is transversal to g1(v1, V 1) for all 1 ≤ l < N .

156

D Monotonicity Formula

The following monotonicity formula is from Huisken [35], and a good reference isthe book from Ecker [18]. We define the m−dimensional backward heat kernelfor x ∈ Rn and t < 0 by

%(x, t) := (−4πt)−m/2

exp

(− |x|

2

(−4t)

)and the backward heat kernel centered at (x0, t0) for x0 ∈ Rn and t < t0 by

%(x0,t0)(x, t) := %(x− x0, t− t0).

Theorem D.1. Let M be a closed m−dimensional manifold i.e. compact andwithout boundary. Let F : M × [0, T∗)→ Rn be a smooth mean curvature flow.Let x0 ∈ Rn and t0 ∈ [0, T∗], and assume that there holds∫

M

%(x0,t0)(x, t) dµt <∞,

for all 0 ≤ t < t0, where x := F (·, t). Then there holds

d

dt

∫M

%(x0,t0) dµt = −∫M

%(x0,t0)

∣∣∣∣H +(x− x0)⊥

2(t0 − t)

∣∣∣∣2 dµt, (181)

for all t ∈ (0, t0).

Proof. We may assume that x0 = 0 and t0 = 0, and calculate

d

dt

∫M

%(F (p, t), t) dµt(p) =

∫M

%t +⟨∇%⊥, H

⟩− %H2 dµt

=

∫M

%t − %∣∣∣∣H − ∇%⊥%

∣∣∣∣2 − ⟨∇%⊥, H⟩+|∇%⊥|2

%dµt.

Since M is closed, by the divergence theorem, there holds∫M

divM (X) dµ =

∫M

divM (X‖)− 〈X,H〉 dµ = −∫M

〈X,H〉 dµ.

With the vector field X = ∇% this implies

d

dt

∫M

% dµt =

∫M

−%∣∣∣∣H − ∇%⊥%

∣∣∣∣2 + %t + divg(∇%) +|∇%⊥|2

%dµt.

By the following calculation

%t =m%

2(−t)− %|x|2

4(−t),

∇% = − %x

2(−t),

divM (∇%) = − m%

2(−t)+%|x‖|2

4(−t)2,

|∇%⊥|2

%=

%|x⊥|2

4(−t)2,

we see that %t + divMt(∇%) + |∇%⊥|2% = 0 and ∇%

⊥

% = − x⊥

2(−t) , and the theorem

follows.

157

The monotonically non-increasing quantity

ψx0,t0(t) :=

∫M

%x0,t0 dµt,

is called the Gaussian density ratio. An immediate consequence of the mono-tonicity formula is that the limit

ψ(x0, t0) := limtt0

ψx0,t0(t),

exists. This limit is called Gaussian density, or sometimes heat density.If the point (x0, t0) is lies in the space time track of the flow, then there

holds the lower bound ψ(x0, t0) ≥ 1. Since the Gaussian density is upper semicontinuous, this holds also for reachable points of the flow, i.e. limit points(xi, ti)→ (x, t), where (xi, ti) lies in the space time track.

If the manifold M is compact, we can also define the maximal Gaussiandensity ratio,

ψt0(t) := supx0∈Rn

ψx0,t0(t),

and the maximal Gaussian density

ψ(t) := limtt0

ψt0(t).

An nice application of the monotonicity formula is the following characteriza-tion. The self similar shrinking solutions, which disappear at the time t = t0 atthe point x0, are characterized by the property that the non-increasing quantityψx0,t0(t) = ψ(x0, t0) is constant for all time t < t0.

We can compare the Gaussian density with the density which arises from thearea ratio. Let g(·, t) := F (·, t)∗δRn be the pullback metric on M , which inducesthe measures µt :=

√det g(·, t) on M . We define the push forward measure

νt := F (·, t)∗µt on Rn. If F (·, t) is an embedding, this measure coincide withHmbF (M, t), but for immersion it takes care of the multiplicity. Hence, generalthere holds HmbF (M, t) ≤ νt. We define the m−dimensional area ratio

Θ(x0, r, t) :=νt(Br(x0))

ωmrm,

where ωm denotes the volume of the unit ball in Rm. We define them−dimensionaldensity

Θ(x0, t) := limr0

Θ(x0, r, t).

If the manifold M is compact, we define the maximal area ratio,

Θ(t) := supr>0

supx0∈Rn

Θ(x0, r, t).

The monotonicity formula implies the following upper bound of the area ratio.

Proposition D.2. Let F : Mm × [0, T∗) → Rn be a smooth mean curvatureflow. If 0 ≤ t0−r2

0 < t0 ≤ T∗, then for all r ∈ (0, r0/√

2(1 + 2m)), and x0 ∈ Rnthere holds

supt∈(t0−r2,t0)

Θ(x0, r, t) ≤ C(m) Θ(x0, r0, t0 − r20/2(1 + 2m)),

where C(m) := 8e1/4(16π(1 + 2m))m/2.

For a proof see Ecker [18], Proposition 4.24.

158

References

[1] U. Abresch & J. Langer, The normalized curve shorting flow and homotheticsolutions, J. Diff. Geom. 23 (1986), 175-196.

[2] F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35(1986), no. 3, 451-547.

[3] L. Ambrosio & H.M. Soner, A level set approach to the evolution of surfacesof any codimension, J. Diff. Geom. 43 (1996), 693-737.

[4] S. Angenent, The zero set of a solution of a parabolic equation, J. reineangew. Math. 390 (1988), 79-96.

[5] S. Angenent, Parabolic equations for curves on surfaces, Part I. Curveswith p-integrable curvature, Ann. of Math. 132 (1990), 451-483.

[6] S. Angenent, Parabolic equations for curves on surfaces, Part II. Intersec-tions, blow-up and generalized solutions, Ann. of Math. 133 (1991), 171-215.

[7] S. J. Altschuler, Singularities of the curve shortening flow for space curves,J. Diff. Geom. 34 (1991), 491-514.

[8] S. J. Altschuler & M. A. Grayson, Shortening space curves and flow throughsingularities, J. Diff. Geom. 35 (1992), 283-298.

[9] K. A. Brakke, The motion of a surface by its mean curvature, Math NotesPrinceton, NJ, Princeton University Press, 1978.

[10] Y.G. Chen, Y. Giga & S. Goto, Uniqueness and existence of viscosity so-lutions of generalized mean curvature flow equations, J. Diff. Geom. 33(1991), 749-786.

[11] K. Deckelnick, Weak solutions of the curve shortening flow, Calc. Var. 5(1997), 489-510.

[12] K. Deckelnick, Partial regularity of weak solutions for the curve shorteningflow, Manuscripta Math. 98 (1999), 265-274.

[13] D.M. DeTurck, Deforming metrics in the direction of their Ricci tensor, J.Diff. Geom. 18 (1983), no. 1, 157-162.

[14] U. Dierkes & S. Hildebrandt & A. Kuster & O. Wohlrab, Minimal SurfacesI/II, Springer, Grundlehre der math 295/296.

[15] J. Douglas, The problem of Plateau for two contours, J. Math. Phys. 10(1931), 315-359.

[16] J. Douglas, Minimal surfaces of higher topological structure, Ann. Math.40 (1939), 205-298.

[17] K. Ecker, On regularity for mean curvature flow of hypersurfaces, Calc.Var. 3, (1995), 107-126.

[18] K. Ecker, Regularity theory for mean curvature flow, Birkhauser Bosten2004.

159

[19] K. Ecker, Partial regularity at the first singular time for hypersurfaces evolv-ing by mean curvature, Math. Ann. 356 (2013) 217-240.

[20] K. Ecker & G. Huisken, Interior estimates for hypersurfaces moving bymean curvature, Invent. Math. 105 (1991), no. 3, 547-569.

[21] K. Ecker & G. Huisken, Mean curvature evolution of entire graphs, Annalsof Mathematics, 130 (1989), 453-471.

[22] L.C. Evans & J. Spruck, Motion of level sets by mean curvature I, J. Diff.Geom. 33 (1991), 635-681.

[23] L.C. Evans & J. Spruck, Motion of level sets by mean curvature II, Trans.Amer. Math. Soc. 330 (1992), no. 1. 321-332.

[24] W. Fenchel, Uber Krummung und Windung geschlossener Raumkurven,Math. Ann. 101 (1929), no. 1, 238252.

[25] M. Gage, Curve shortening makes convex curve circular, Invent. Math. 76(1984), 357-364.

[26] M. Gage & R. S. Hamilton, The heat equation shrinking convex planecurves, J. Diff. Geom. 23 (1986), 69-96.

[27] D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of sec-ond order, Classics in Mathematics, Springer.

[28] M. A. Grayson, The heat equation shrinks embedded plane curves to roundpoints, J. Diff. Geom. 26 (1987), 285-314.

[29] M. A. Grayson, Shortening embedded curves, Annals of Mathematics, 129(1989), 71-111.

[30] V. Guillemin & A. Pollack, Differential topology, Prentice-Hall, 1937.

[31] R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in theplane, Modern methods in Complex analysis, Princeton Univ. press (1992),201-222.

[32] J. Hattenschweiler, Mean curvature flow of networks with triple junctionsin the plane, Diplomarbeit ETH Zurich 2007.

[33] X. Han & J. Sun An ε-regularity theorem for the mean curvature flow,arXiv:1102.4800 (2011).

[34] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J.Diff. Geom. 20 (1984), 237-266.

[35] G. Huisken, Asymptotic behavior for singularities of the mean curvatureflow, J. Diff. Geom. 31 (1990), 285-299.

[36] G. Huisken, A distance comparison principle for evolving curves, Asian J.Math. 2 (1998), no. 1, 127-133.

[37] G. Huisken & A. Polden, Geometric evolution equations for hypersurfaces,Calculus of variations and geometric evolution problems (Cetraro, 1996)Springer 1999, 45-84.

160

[38] T. Ilmanen, Elliptic regularization and partial regularity for motion by meancurvature, Mem. Amer. Math. Soc., vol. 108(520), AMS, 1994.

[39] T. Ilmanen, Singularities of mean curvature flow of surfaces, 1995.

[40] T. Ilmanen, Lectures on mean curvature flow and related equations, 1998.

[41] N.V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces,Garduate Studies in Mathematics, Vol. 12, AMS 1996.

[42] O.A. Ladyzhenskaya, V.A. Solonnikov & N.N. Ural’ceva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Mono-graphs, 23, AMS 1968.

[43] G. M. Lieberman, Second order parabolic differential equations, World Sci-entific Publishing, 1998.

[44] C. Mantegazza, Lecture notes on mean curvature flow, Progress in Mathe-matics Vol. 290, Birkhauser 2010.

[45] J. Moser, A Harnack inequality for parabolic differential equations, Comm.on Pure and Appl. Math. 17 (1964), no 1, 101-134.

[46] J. W. Morgan & G. Tian, Ricci Flow and the Poincare conjecture,arXiv:math/0607607v2

[47] J.C.C. Nitsche, Vorlesung uber Minimalflachen, Springer, Grundlehre dermath 199.

[48] H.M. Soner, Motion of a set by the curvature of its boundary, J. Diff. Eqs.101 (1993), no. 2, 313-372.

[49] G. Perelman, Finite extinction time for the solutions to the Ricci flow oncertain three manifolds, arXiv:math/0307245v1

[50] M. Struwe, A Morse theory for annulus-type minimal surfaces, J. reineangew. Math. 368 (1984), 1-27.

[51] B. White, A local regularity theorem for the mean curvature flow, Ann.Math. 161 (2005), 1487-1519.

[52] Yun Yan Yang & Xiao Xiang Jiao, Curve shortening flow in arbitrary di-mensional Euclidean space, Acta Math Sinica. 21 (2005), 715-722.

[53] X. Zhu, Lectures on mean curvature flows, AMS Studies in Advanced Math,Vol 32 (2002).

161

Notation Index

F : Mm → Pn immersion of Riemannian manifolds, page 7x(u, t) curve shortening flow in Rn page 12k = |k|N vector valid curvature of x, page 12(T,N,B1, ..., Bn−2) Frenet-Serret frame of a space curve x, page 13T∗ first singular time of the smooth flow page 12µt Radon measure on S1 induced by x(·, t), page 14K(t) total absolute curvature of x(·, t), page 15Kp(t) Lp norm of the curvature, page 31Lipε(x), αε(x) Lipschitz ε−graph constant page 43κt = |k(·, t)|dµt Radon measure on S1 page 67D(x), D0(x) disk-size of an arc resp. closed curve page 86D(x, y), D0(x, y) strip pseudo-distances page 86x ? y concatenation of curves, page 88L(x) length of the Lipschitz map x page 86A(F ) area of the Lipschitz map F page 86distx, distF intrinsic distance by the pull back metricT ∗ extinction time of the weak flow page 122U ⊂ (0, T ) open full measure set of weak solution page 122R,S,Rp, Sp regular, singular, past-regular, past-singular points, page 124R,S,Rp,Sp regular, singular, past-regular, past-singular times, page 124X(u, t) space time parametrization, i.e. (x(u, t), t) page 137Γ space time track, i.e. the image of X page 137Σp past-singular space-time points page 137x+, x− left- and right limit of a weak solution page 133µ+t , µ

−t left- and right limit of the Radon measure page 133

Rp(t), Sp(t) past-regular and past-singular points at time t page 143Σp(t) past-singular space points at time t page 143%(x0,t0)(x, t) backward heat kernel page 157ψx0,t0(t) Gaussian density ratio page 157ψ(x, t) Gaussian density page 157

162

Acknowledgments

Ich bedanke mich herzlich bei Tom Ilmanen fur die gute Betreuung, seinenaturliche Art welche die Zusammenarbeit sehr angenehm, kreative und pro-duktive machte und seine vielen Tipps, die mich nicht nur mathematisch weiterbrachten. Zudem danke ich Ihm fur die schier unendliche Geduld und Ausdauer,welche ich auf die Probe gestellt habe.

Dominique Aberlin danke ich fur das Korrektur lesen. Seine zu mir unter-schiedliche Sichtweise brachten mir fachlich immer wieder neue Zugange undfuhrten im allgemeinen zu guten Diskussionen. Ruben Jakob danke ich fursein Interesse an meiner Arbeit, seine aufopfernde Hilfebereitschaft und dasAuffinden von 1001 Fehlern. Ein herzliches Dankeschon geht an das gesamteD-Math und insbesondere meinen Buro Kollegen Felix Hensel, Kathrin Naefund Michael Siepmann, welche mir die Arbeitszeit versußten.

Nicht zuletzt danke ich meiner Familie, meiner Mutter und meiner SchwesterMonica fur ihre Liebe und Unterstutzung.

163

Curriculum Vitae

Personal DataName Jorg HattenschweilerDate of birth November 30, 1974Place of birth Dielsdorf SwitzerlandCitizenship Swiss

Education2007-15 Doctoral studies in Mathematics under the supervision of

Prof. Tom Ilmanen, ETH Zurich, Switzerland2002-06 Diploma studies in Mathematics, ETH Zurich, Switzerland

Thesis: Mean Curvature Flow of Networks withTriple Junctions in the Plane

1996-99 FH Horw Hochschule fur Technik + Architektur Luzern1991-95 Lehre Schupbach Engineering AG, Glattbrugg1981-90 Primary and Secondary School Schleinikon, Niederweningen

Employment History2007-14 Assistant at ETH Zurich for the Department Mathematics2005-06 Teaching aid at the ETH for the Department of Mathematics2000-02 Amstein & Walthert AG, Oerlikon1991-96 Schupbach Engineering AG, Glattbrugg

164

· abstract we study the mean curvature ow of closed curves in an euclidean space and de ne weak...

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