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AN INTRODUCTION TO THE MEAN CURVATUREFLOW

FRANCISCO MARTIN AND JESUS PEREZ

Abstract. The purpose of these notes is to provide an introduc-tion to those who want to learn more about geometric evolutionproblems for hypersurfaces and especially those related to curva-ture flow. These diffusion problems lead to interesting systems ofnonlinear partial differential equations and provide the appropriatemathematical modeling of physical processes.

Contents

1. Introduction 2

2. Existence y uniqueness 12

3. Evolution of the Geometry by the Mean Curvature Flow 18

4. A comparison principle for parabolic PDE’s 31

5. Graphical submanifolds. Comparison Principle andConsequences 34

6. Area Estimates and Monotonicity Formulas 52

7. Some Remarks About Singularities 72

References 74

Date: July 18, 2014.1991 Mathematics Subject Classification. Primary 53C44,53C21,53C42.Key words and phrases. Mean curvature flow, singularities, monotonicity for-

mula, area estimates, comparison principle.Authors are partially supported by MICINN-FEDER grant no. MTM2011-

22547.1

2 FRANCISCO MARTIN AND JESUS PEREZ

1. Introduction

Mean Curvature Flow is an exciting and already classical mathemati-cal research field. It is situated at the crossroads of several scientific dis-ciplines: Geometric Analysis, Geometric Measure Theory, PDE’s The-ory, Differential Topology, Mathematical Physics, Image Processing,Computer-aided Design, among other. The purpose of these notes is toprovide an introduction to those who want to learn more about thesegeometric evolution problems for curves and surfaces and especiallycurvature flow problems. They lead to interesting systems of nonlinearpartial differential equations and provide the appropriate mathematicalmodeling of physical processes such as material interface propagation,fluid free boundary motion, crystal growth,...

In Physics, diffusion is known as a process which equilibrates spatialvariations in concentration. If we consider a initial concentration u0 ona domain Ω ⊆ R2 and seek solutions of the linear heat equation

(1.1)∂

∂tu−∆u = 0,

with initial data u0 and natural boundary conditions on ∂Ω, we obtaina successively smoothed concentrations utt>0. When Ω = R2, thesolutions to this parabolic PDE coincides with the convolution of theinitial data with the heat kernel (or Gaussian filter)

Φσ(x) =1

2πσe−|x|

2/σ2

with standard deviation sigma, i.e., ut2/2 = Φt ∗ u0 (see Remark 6.18.)In general, derivatives of ut are bounded for t > 0 in terms of boundson u0. It follows that, even if you start with a heat distribution whichis discontinuous, it immediately becomes smooth. Moreover, solutionsconverge smoothly (in C∞) to constants as t→∞ (eventual simplicity).

The heat equation has some surprising properties which carry over tomuch more general parabolic equations.

The Maximum Principle: : At a point where ut attains a max-imum in space (that is, in Ω), the second derivatives in each di-rection are non-positive. By the heat equation, the time deriva-tive is non-positive. It follows that the maximum temperature,umax(t) = supx∈Ω u(x, t), does not increase as time passes.

Gradient Flow: A further useful property which holds for manybut not all heat-type equations is the gradient property: The

MEAN CURVATURE FLOW 3

heat equation is the flow of steepest decrease of the DirichletEnergy:

E(u) =1

2

∫Ω

|(Du)(x)|2dx.

Figure 1. A surface moving by mean curvature.

If we are interested in the smoothing of perturbed surface geometries,it make sense to think in analogues strategies. So, the source of inspi-ration diffused throughout everything that follows is the classical heatequation (1.1).

The geometrical counterpart of the Euclidean Laplace operator ∆ ona smooth surface M2 ⊂ R3 (or more generally, a hypersurface Mn ⊂Rn+1) is the Laplace-Beltrami operator, that we will denote as ∆M .


Thus, we obtain the geometric diffusion equation

(1.2)∂

∂tx = ∆Mt x,

for the coordinates x of the corresponding family of surfaces Mtt∈[0,T ).

A classical formula by Weierstraß (see [DHKW92], for instance) saysthat, given an orientable1 (hyper)surface in Euclidean space, one has:

∆Mt x = ~H,

where ~H means the mean curvature vector. This means that (1.2) canbe written as:

(1.3)∂

∂tx(p, t) = ~H(p, t)

The mean curvature is known to be the first variation of the areafunctional M 7→

∫Mdµ (see [DHKW92,CM11,MIP12].) We will obtain

for the Area(Ω(t)) of a relatively compact Ω(t) ⊂Mt that

d

dt(Area(Ω(t)) = −

∫Ω(t)

| ~H|2dµt.

In other words, we get that the mean curvature flow is the correspond-ing gradient flow for the area functional:

The Mean Curvature Flow is the flow of steepest decreaseof surface area.

Moreover, we also have a nice maximum principle for this particulardiffusion equation.

Theorem (Maximum/Comparison principle). If two properly immersedhypersurfaces of Rn+1 are initially disjoint, they remain so. Further-more, embedded hypersurfaces remain embedded.

In this line of result, we would like to point out that:

• If the initial hypersurface M is convex (i.e., all the geodesic cur-vatures are positive, or equivalently M bounds a convex regionof Rn+1), then Mt is convex, for any t.• If M is mean convex (H > 0), then Mt is also mean convex, for

any t.

1Throughout these notes we shall always assume that the hypersurfaces of Rn+1

are orientable.


Moreover, mean curvature flow has a property which is similar to theeventual simplicity for the solutions of the heat equation. This resultwas proved by Huisken and asserts:

Theorem. Convex, embedded, compact hypersurfaces converge to pointsp ∈ Rn+1. After rescaling to keep the area constant, they convergesmoothly to round spheres.

There is a rather general procedure for producing heat-like curva-ture flows. In general, we wish to evolve hypersurfaces Mn in Rn+1

(or in a complete, Riemannian, (n + 1)-dimensional manifold). Thenany (smooth) symmetric function f of n variables, which is monotoneincreasing in each variable, determines a suitable speed function:

F (p, t) := f(k1(p, t), . . . , kn(p, t));

where ki, i = 1, . . . , n, represent the principal curvatures of Mt. Thisyields a general class of curvature flows:

(1.4)∂

∂tx(p, t) = F (p, t) · ν(p, t),

where ν(·, t) is the Gauß map of Mt. Some of the most interestingexamples are:

(a) Mean Curvature Flow: f(x1, . . . , xn) =n∑i=1

xi, (F = H).

(b) Harmonic Mean Curvature Flow: f(x1, . . . , xn) =

(n∑i=1

1

xi

)−1

.

(c) Gauß curvature flow: f(x1, . . . , xn) =n∏i=1

xi, (F = K).

(d) Inverse Mean Curvature Flow: f(x1, . . . , xn) = − 1∑ni=1 xi

, (F =

−H−1).

Applications of the mean curvature flow (and its variants: harmonicmean curvature flow, inverse mean curvature flow,...) are numerous andcover various aspects of Mathematics, Physics and Computing. In thefollowing paragraphs we will briefly describe some of these applications,with particular emphasis on two of them. The inverse mean curvatureflow was used by Huisken and Ilmanen to prove the Riemann Penroseinequality [HI01]. Similarly, Andrews got an alternative proof of thetopological version of the sphere theorem [And94] making use of theharmonic mean curvature flow.


1.1. Riemannian Penrose Inequality. The Riemannian Penrose in-equality is a special case of the unsettled Penrose Conjecture. In aseminal paper [Pen73] (see also [Pen82]), in which he proposed the cel-ebrated cosmic censorhip conjecture, R. Penrose also proposed arelated inequality, which today is know as “Penrose Inequality”. Theinequality is derived from cosmic censorship by using a heuristic argu-ment relying on Hawking’s Area Theorem [HE73]. Consider a space-time satisfying the so called dominant energy condition (DEC),which contains an asymptotically flat Cauchy surface with ADM massm (see definition below), and containing an event horizon (roughly, thearea of a black hole) of area A = 4πr2, which undergoes gravitationalcollapse and settles to a Kerr-Newman solution of mass m∞ and arearadius r∞. Physical arguments imply that the ADM mass of the finalstate m∞ is no greater than m (no new mass appear, even though ra-diation may imply some loss of mass), then the area radius r∞ is noless than r, and the final state must satisfy

m∞ ≥1

2r∞.

The evolution of black holes (assuming that it is deterministic, i.e., nonaked singularity appears) implies that the area of its event horizonmust increase, so it must have been the case that

m ≥ 1

2r,

also at the beginning of the evolution.

A counterexample to the Penrose inequality would therefore suggestsdata which leads under the Einstein evolution to naked singularities,and a proof of the Penrose inequality may be viewed as evidence insupport of the cosmic censorship. The event horizon is indiscerniblein the original slice without knowing the full evolution, however onemay, without disturbing this inequality, replace the event horizon bythe (possible smaller) apparent horizon, the boundary of the regionadmitting trapped surfaces. The inequality is even more simple in thetime-symmetric case, in which the apparent horizon coincides with theoutermost minimal surface, and the dominant energy condition reducesto the condition of nonnegative scalar curvature. This leads to theRiemannian Penrose inequality: the ADM mass m and the area radiusr of the outermost minimal surface in an asymptotically at 3-manifoldof nonnegative scalar curvature, satisfy

m ≥ r

2=

√A

16π,


and the equality holds if and only if the manifold is isometric to thecanonical slice of the Schwarzschild spacetime. Note that this charac-terizes the canonical slice of Schwarzschild as the unique minimizer ofm among all such 3-manifolds admitting an outermost horizon of areaA.

For a more precise explanation about the physical interpretation ofthe inequality we recommend [MS13]. In these notes, we will focus onthe mathematical aspects of the Riemannian Penrose inequality andhow the ICMF has been a key tool in their demonstration.

Consider (N3, g = (gij)) a Riemannian 3-manifold. Assume N isasymptotically flat which means that:

(1) N is realized by an open set which is diffeomorphic to R3 \K;K compact,

(2) |gij − δij| ≤C

|x|, as |x| → ∞;

(3)

∣∣∣∣∂gij∂xk

∣∣∣∣ ≤ C

|x|2, as |x| → ∞.

(4) We also assume

Ric ≥ − C

|x|2· g.

In this setting we define

Definition 1.1 (Arnowitt-Deser-Misner (ADM) mass). The total en-ergy, or ADM mass, of the end is defined by a flux integral through thesphere at infinity:

m := limr→+∞

1

16π

∑i,j

∫S2(0,r)

(∂gij∂xj− ∂gij∂xi

)· nj dµ,

where n represents the “outward” pointing Gauß map of the Euclideansphere S2(0, r).

Although this flux is defined using local coordinates, it is globalinvariant of the end.

Theorem 1.2 (Riemannian Penrose Inequality [HI01]). Let N be acomplete, connected 3-manifold (with boundary.) Suppose that

(a) N is asymptotically flat, with ADM mass m,(b) N has nonnegative scalar curvature,


(c) N has compact boundary which consists of minimal surfaces, andN contains no other compact minimal surfaces.

Then

m ≥√

Area(M)

16π,

where M is any connected component of ∂N . Moreover, equality holdsiff N is isometric to one-half of the spatial Schwarzschild manifold.

The spatial Schwarzschild manifold is (R3 − (0, 0, 0), g) where

g :=

(1 +

m

2|x|

)4

· g0,

and g0 represents the Euclidean metric of R3.

• It possesses an inversive isometry fixing S2(0,m/2), which is anarea minimizing sphere of area 16πm2.• The manifold of the Riemannian Penrose inequality is R3 −B(0,m/2).

Huisken-Ilmanen’s proof of the Riemannian Penrose inequal-ity. Huisken and Ilmanen proved the inequality, including the rigiditypart, in [HI01] by using the inverse mean curvature flow, an ap-proach proposed by Jang and Wald [JW77]. In this introduction, wewould like to give a rough idea of the structure of their proof. A classi-cal solution of the Inverse Mean Curvature Flow (IMCF from now on)is a smooth family of (hyper)surfaces F : M × [0, T ] → N, satisfyingthe equation

(1.5)∂

∂tF = − ν

H,

where ν represents the Gauß map of Mt := F (M, t) and 0 < H(·, t)is its mean curvature.

Without extra geometric hypotheses, the mean curvature could de-velop a zero and then the equation would present a singularity. Thatwas the reason because Huisken and Ilmanen introduced a level-set for-mulation of (1.5), where the evolving surfaces are given as level sets ofa scalar function φ:

Mt = ∂ (x ∈ N / φ(x) < t) ,so (1.6) is replaced by the degenerate elliptic equation


(1.6) divN

(Dφ

|Dφ|

)= |Dφ|

They were able to overcome the problem that the evolving surface canbecome singular before reaching infinity by formulating and analysing asuitable weak notion of the solution of (1.6). These weak solutions arelocally Lipschitz continuous functions and the treatment was inspiredin a work of Evans and Spruck on the MCF [ES91].

It appears that neither (1.5) is a gradient flow nor (1.6) is an Euler-Langrange equation. The idea of these two authors consists of freezing|Dφ| in the right-hand side of (1.6) and consider (1.6) as the Euler-Lagrange equation of the functional:

Jφ(ψ) = JKφ (ψ) :=

∫K

(|Dψ|+ ψ|Dφ|) d µ,

where K is a compact subset of N .

Definition 1.3 (Weak solution). Let φ a locally Lipschitz function onthe open set Ω ⊆ N . Then we say that φ is a weakk solution of (1.6)on Ω provided

JKφ (φ) ≤ JKφ (ψ),

for all ψ locally Lipschitz and such that ψ 6= φ ⊂⊂ Ω, where we areintegrating over any compact ψ 6= φ ⊆ K ⊂ Ω.

In order to understand the value of the above definition in the solutionof our problem, we need to introduce a new concept; the Hawking mass.Given a compact surface M in N , the Hawking mass is defined by:

mH(M) :=

√Area(M)

(16π)3

(16π −

∫M

H2dµ

).

Hawking already noticed that mH approaches the ADM mass for largecoordinate spheres.

If M is minimal then mH(M) is precisely

√Area(M)

16π. Moreover, the

Hawking mass has an especially nice behavior respect to the inverseMCF:

I. Geroch Monotonicity Formula. Geroch [Ger73] introducedthe IMCF and realized that the mass mH of a family of surfaces


evolving by the inverse MCF is monotone nondecreasing, pro-vided that the surface is connected and the scalar curvature of Nis nonnegative.

II. One of the main achievements of [HI01] consists of proving thatGeroch Monotonicity Formula also works for the Huisken-Ilmanenweak solutions of the inverse MCF, even in the presence of jumps.

III. The derivative vanishes precisely on standard expanding spheresin flat 3-space and Schwarzschild example.

Taking these properties into account the proof of the case of a connectedhorizon (M = ∂N connected) works as follows. We move M by theIMCF, obtaining a family of compact surfaces Mt which collapses atthe point of infinity. By the monotonicity formula, we know that

mH(Mt) ≥ mH(M) =

√Area(M)

16π,

(recall that M is minimal.) For t big enough, we have that mh(Mt)approximates the ADM mass m, so

m ≈ mH(Mt) ≥√

Area(M)

16π.

Moreover, the equality holds iff mH(Mt) is constant along the flow, i.e.its derivative vanishes. According to Property III, this only happensfor standard expanding spheres in Schwarzschild’s example.

Finally, we would like to mention that the inequality was proven in fullgenerality (non-connected horizon) by Bray [Bra01] using a conformalflow of the initial Riemannian metric, and the positive mass theorem[SY79].

1.2. The Sphere Theorem. It is known that if a manifold is simplyconnected and has constant positive sectional curvatures, then it is asphere with the standard Riemannian metric.

• In the 1940’s, Heinrich Hopf asked whether we can also wig-gle the geometry a little, instead only requiring that the sec-tional curvatures be close to some constant. Let’s say 1 − ε <K ≤ 1.• In 1951 Rauch [Rau51] proved a simply connected manifold with

curvature in [3/4, 1] is homeomorphic to a sphere.• At the beginning of the 1960’s, Berger [Ber60] and Klingen-

berg [Kli61] confirmed the conjecture, with the optimal value


of ε : A simply connected Riemannian manifold with sectionalcurvatures in the interval (1/4, 1] is homeomorphic to a sphere.

If the value 1/4 is allowed, there are counterexamples. Ac-tually, any compact symmetric space of rank 1 admits a metricwhose sectional curvatures lie in the interval [1, 4]. The list ofthese spaces includes the following examples:

– The complex projective space CPk, for 2k ≥ 4.– The quaternionic projective space HPk, for 4k ≥ 8.– The projective plane over the octonions (dimension 16)

It remained an open conjecture for over 50 years that the conclusion ofhomeomorphism should be improvable to diffemorphism. The problemwas solved in the affirmative by S. Brendle and R. Schoen in [BS09]using the Ricci Flow.

However, as we mentioned before, the sphere theorem was also provedby B. Andrews using curvature flows.

Idea of the proof: Using the pinching assumption, it is not difficultto construct a large disk D(p, r) in M whose boundary is smooth andconvex in the “outwards” direction.

We would like to flow this boundary in the outwards direction to apoint via a suitable curvature flow. This would demonstrate that themanifold is formed from gluing two disks together along their bound-aries, and hence is a sphere.

We know that the mean curvature flow doesn’t work, but there’s atleast one flow speed that makes the job; namely, the harmonic meancurvature:

∂

∂tx = f · ν, f(k1, . . . , kn) =

(n∑i=1

1

ki

)−1

.

Note that the conclusion in this case is stronger than homeomor-phism: The manifold is diffeomorphic to a twisted sphere (two disksglued by a diffeomorphism along their boundary). But this is stillslightly weaker than diffeomorphism.

1.3. Image processing. These ‘smoothing” properties that we men-tioned of mean curvature flow make it an ideal tool in image processing,computer aided geometric design and computer graphics. Here, is-sues are fairing, modeling, deformation, and motion. Constructive and


more explicit approaches based for instance on splines are nowadaysalready classical tools. More recently geometric evolution problemsand variational approaches have entered this research field as well andhave turned out to be powerful tools. For those readers interested inthe applications of curvature flows in image processing we recommend[CDR03].

Since our background is closer to Geometric Analysis, we have mainlyused the monographs [Eck04], [Man11] and [RS10] in the elaborationof these notes. For readers who are more familiar with the languageand techniques of Geometric Measure Theory, we recommend [Bra78]and [Ilm95].

These notes correspond to the contents of a mini-course given bythe first author at the Program on Geometry and Physics, Granada2014. The authors are extremely grateful to all participants in thisprogram who have sent us corrections and suggestions that have helpedto improve these notes. In that sense, we feel especially indebted toMiguel Sanchez for his valuable comments.

2. Existence y uniqueness

Along this section M will represent a n-dimensional submanifold ofRn+1. Although the most part of the results are also valid in a moregeneral setting, we will restrict our attention to the study of hyper-surfaces in Euclidean space.

Definition 2.1. We say that M moves by the mean curvature if thereexists a smooth family of immersions F : M × [0, T )→ Rn+1 such that

• F (·, 0) is the original immersion of M in Rn+1;• For each p in M and for each t in [0, T ) one has

∂

∂tF (p, t) = ~H(p, t),

where ~H(p, t) is the mean curvature vector of F (M, t) at F (p, t).

Remark 2.2. We shall introduce the following notation. Given a mapF : M × [0, T ) → Rn+1 as in the previous definition, we will writeFt := F (·, t). Thus, we will use the same notation for all the elements


associated to the smooth family of immersions Ftt∈[0,T ) like, for in-

stance Mt := F (M, t), the mean curvature vector ~Ht(p) := ~H(p, t), etc.For simplicity in the computations, we will often suppress the subscriptt when no confusion is possible.

We will start with some local computations. Let (U, x1, x2, . . . , xn)be local co-ordenates around p ∈ U ⊂ Mn. We have F : (Mn, g) →(Rn+c, g) where g = 〈·, ·〉 is the usual scalar product in Rn+c and g =dF ∗(g). So

gij = g(∂

∂xi,∂

∂xj) = g

(dF

(∂

∂xi

), dF

(∂

∂xj

))=

⟨∂F

∂xi,∂F

∂xj

⟩,

where we are writing dF(∂∂xi

)= ∂F

∂xi.2

Recall that we can locally identify Mn with F (Mn). In this way,we can say that ∂F

∂xiis a local vector field on Rn+k which extends the

smooth field on Mn given by ∂∂xi

.

On the other hand, the Levi-Civita connection ∇ in the Euclideanspace Rn+c is the standard flat connection in Rn+c, i.e., given X, Y ∈X(Rn+c) one has ∇X Y = DX Y , where DX Y means the directionalderivative of Y alongX. In what follows it will appear ∇ ∂F

∂xi

∂F∂xj

, that

we will denote as ∂2F∂xi∂xj

. It is an abuse of notation whose justification

is as follows:

∇ ∂F∂xi

∂F

∂xj= D ∂F

∂xi

∂F

∂xj= D ∂

∂xi

∂F

∂xj=

∂2F

∂xi∂xj,

where the second equality has taken into account the identificationbetween the fields ∂

∂xiand ∂F

∂xigiven by F . We have also used that

∇X Y = DX Y where D means the standard directional derivative inRn+c.

Recall that the mean curvature vector can be computed in terms ofthe connection as follows3:

~H =

(gij∇ ∂F

∂xi

∂F

∂xj

)⊥.

2We are considering dF as a map from X(U) into X(F (U)), in such a way thatdF(

∂∂xi

)can be seen as a local field on Rn+c.

3Given a vector v ∈ Rn+c and p ∈ M we can decompose v = v> + v⊥ wherev> ∈ TpM and v⊥ ∈ (TpM)⊥


And we can go further getting that:(gij∇ ∂F

∂xi

∂F

∂xj

)⊥= gij∇ ∂F

∂xi

∂F

∂xj−(gij∇ ∂F

∂xi

∂F

∂xj

)>=

= gij∇ ∂F∂xi

∂F

∂xj−⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gkl

∂F

∂xl.

Proof. Indeed, we only have to check that(gij∇ ∂F

∂xi

∂F

∂xj

)>=

⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gkl

∂F

∂xl.

To do this, fix m ∈ 1, 2, . . . , n, then one has⟨⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gkl

∂F

∂xl,∂F

∂xm

⟩=

⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gkl⟨∂F

∂xl,∂F

∂xm

⟩=

=

⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gklglm =

⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩δkm =

=

⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xm

⟩=

⟨(gij∇ ∂F

∂xi

∂F

∂xj

)>,∂F

∂xm

⟩.

Using the non-degenerancy of the metric 〈·, ·〉 we complete the proof.

Hence, the Mean Curvature Flow equation ∂∂tF (p, t) = ~H(p, t) can be

written in this new form:

∂

∂tF (p) = gij∇ ∂F

∂xi

∂F

∂xj−⟨gij∇ ∂F

∂xi

∂F

∂xj,∂F

∂xk

⟩gkl

∂F

∂xl,

taking into account that ∇ ∂F∂xi

∂F∂xj

= ∂2F∂xi∂xj

we obtain

∂

∂tF (p) = gij

∂2F

∂xi∂xj−⟨gij

∂2F

∂xi∂xj,∂F

∂xk

⟩gkl

∂F

∂xl.

If we express the previous equation in coordinates we get:

∂

∂tFα(p) = gij

∂2Fα

∂xi∂xj− gijgkl

n+k∑β=1

∂2F β

∂xi∂xj

∂F β

∂xk

∂Fα

∂xl,

where we clearly observe that we are dealing with a non-linear PDE.4

4The second order coefficients are gij , which depend on the map F .


Another way of writing the MCF equation is the following:

~H =

(gij∇ ∂F

∂xi

∂F

∂xj

)⊥= gij∇ ∂F

∂xi

∂F

∂xj−(gij∇ ∂F

∂xi

∂F

∂xj

)>=

= gij(

∂2F

∂xi∂xj−(∇ ∂F

∂xi

∂F

∂xj

)>)= gij

(∂2F

∂xi∂xj−∇ ∂F

∂xi

∂F

∂xj

).(2.1)

At this point, we would like to remind some notions related to the hes-sian and laplacian operators.

Definition 2.3. Let (Mn, g) be a Riemannian manifold and let ∇ beits Levi-Civita connection. The hessian of f ∈ C∞(M) is defined asthe operator ∇2f : X(M) × X(M) → C∞(M) given by ∇2f(X, Y ) :=X(Y (f))− (∇XY )(f) for any X, Y ∈ X(M).

It is well known that ∇2f is symmetric (to prove this we use that∇ is torsion free) and C∞(M)-bilinear.

Definition 2.4. Given (Mn, g) a Riemannian manifold and f ∈ C∞(M)we define the Laplacian of f as ∆f := Trace(∇2f).

Notice that the map F : Mn → Rn+c does not have a well definedLaplacian. However, it makes sense to define the laplacian of each com-ponent Fα. So, we define the laplacian of F as ∆F := (∆F 1,∆F 2, . . . ,∆F n+c).Analogously, we can define also the hessian of F .Bearing this in mind, we have that

∇2F

(∂F

∂xi,∂F

∂xj

)=∂F

∂xi

(∂F

∂xj(F )

)−(∇ ∂F

∂xi

∂F

∂xj

)(F ) =

=∂

∂xi

(∂

∂xj(F )

)−(∇ ∂F

∂xi

∂F

∂xj

)(F ) =

=∂2F

∂xi∂xj−(∇ ∂F

∂xi

∂F

∂xj

)(F )

and so equation (2.1) becomes

~H = gij · (∇2F )

(∂F

∂xi,∂F

∂xj

)= Trace(∇2F ) = ∆F,

Remark 2.5. Recall that ~H depends on the point p ∈ M and theinstant t. In particular, we want to emphasize the temporal dependenceof the mean curvature. Moreover, the intrinsic metric g also depends


on t. For this reason, it is more precise to denote the intrinsic laplacianas ∆g(t)F .

Taking into account the above remark, one has:

~H = ∆g(t)F,

which provides a new way of writing the MCF equation:

(2.2)∂F

∂t= ∆g(t)F.

Using this expression we will prove the existence of the mean curva-ture flow for a small time period in the case of compact manifolds. Inthe demonstration, we will use a technique known as “de Turck trick”.As we shall see, this trick consists of reducing the original problem toa strictly parabolic quasilinear problem.

Theorem 2.6 (Short Existence and Uniqueness). Let M be acompact manifold and F0 : M → Rn+1 a given immersion. There existsa positive constant T > 0 and a unique smooth family of immersionsF (·, t) : M → Rn+1, t ∈ [0, T ), such that

∂

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

F (·, 0) = F0.

Proof. First of all, notice that F0 = F (·, 0) is an immersion, so F (·, t)also would be an immersion for t small enough (see, for instance, [GP74,p. 35].) That means that we only take care about the existence andsolution of the above PDE.

Assume that for some vector field V = vk ∂∂xk

in M (the field V will

be fixed later) we have that the equation:

(2.3)∂F

∂t= ∆g(t)F + vk

∂F

∂xk

has solution for initial data F0, F : M × [0, T ) → Rn+1. We are goingto see that the same happens for the MCF equation with initial dataF0.Indeed, consider a family ϕt : M×[0, T )→M of diffeomorphisms of M .Let Ft(p) := Ft(ϕt(p)) = F (ϕt(p), t), where F is the aforementioned


solution of (2.3) (for now, we are assuming that such a solution exists.)Using the chain rule and (2.3) we compute ∂Ft

∂t(p):

∂Ft∂t

(p) =∂F (ϕt, t)

∂t(p) =

∂F

∂xk(ϕt(p), t)

∂ϕkt∂t

(p) +∂F

∂t(ϕt(p), t) =

=∂F

∂xk(ϕt(p), t)

∂ϕkt∂t

(p) + ∆g(t)F (ϕt(p), t) + vk∂F

∂xk(ϕt(p), t) =

= ∆g(t)F (ϕt(p), t) +∂F

∂xk(ϕt(p), t)

(vk +

∂ϕkt∂t

(p)

).

Hence, to get a solution to the MCF equation it suffices to find a familyϕt such that:

∂ϕt∂t

= −V,

ϕ0 = id .

This is a initial value problem for a system of ODE’s and so we can finda solution. Moreover, taking T > 0 small enough we can assume thatϕt is a diffeomorphism, for any t ∈ [0, T ]. This is due to the fact thatthe initial data is a diffeomorphism (the identity) and the fact that thediffeomorphisms from a compact manifold into itself form a stable class(see again [GP74, p. 35].)Hence, Ft(p) = F (ϕt(p), t) verifies:

∂Ft∂t

(p) = ∆g(t)F (ϕt(p), t) = ∆g(t)Ft(p),

F (p, 0) = F (ϕ0(p), 0) = F (id(p), 0) = F (p, 0) = F0(p),

in other words, it represents a solution of the MCF equation with initialdata F0.

Summarizing, we only have to see that (2.3) has a solution. To dothis we take the vector field V whose coordinates are given by vk :=gij(Γkij − (Γ0)kij), being Γkij the Christoffel symbols of M ((Γ0)kij means


the Christoffel symbol at t = 0.) Then (2.3) becomes

∂F

∂t= ∆g(t)F + vk

∂F

∂xk

= gij(

∂2F

∂xi∂xj−∇ ∂F

∂xi

∂F

∂xj

)+ gij(Γkij − (Γ0)kij)

∂F

∂xk

= gij(

∂2F

∂xi∂xj− Γkij

∂F

∂xk

)+ gij(Γkij − (Γ0)kij)

∂F

∂xk

= gij(

∂2F

∂xi∂xj− (Γ0)kij

∂F

∂xk

).

Thus, this particular choice of V implies that the equation:

∂F

∂t= ∆g(t)F + vk

∂F

∂xk

can be written (in coordinates) as follows:

∂F

∂t= gij

∂2F

∂xi∂xj− gij(Γ0)kij

∂F

∂xk,

which a system of quasilinear parabolic PDE’s because (gij) is a positivedefinite matrix which only depends on the first derivatives of F . Thelocal theory of parabolic PDE’s [Tay96] and the fact that M is compact,gives us the existence and uniqueness of the solution in a short intervalof time [0, T ). This concludes the proof.

3. Evolution of the Geometry by the Mean CurvatureFlow

In this section we study how the usual geometric quantities evolveunder the mean curvature flow.

Remark 3.1 (Notation). From now on, we will denote ∇iX = ∇ ∂∂xi

X

for any X ∈ X(M).For the sake of simplicity, we will often write:

∇if =∂f

∂xifor any function f ∈ C∞(M),

∇iX =∂

∂xiX for any field X ∈ X(M).


Theorem 3.2 (Evolution of the intrinsic geometry). The intrin-sic metric and the volumen form evolve as follows:

(3.1)∂

∂tgij = −2〈 ~H,Aij〉

(3.2)∂

∂t

√det g = −| ~H|2

√det g,

where Aij := II

(∂∂xi, ∂∂xj

)and II(·, ·) denotes the second fundamental

form of the corresponding immersion.

Proof. As all the computations are local, then we can assume that Fis an embedding.Given a vector field X in Rn+c we shall write:

∇iX := ∇ ∂F∂xi

X, ∇tX := ∇ ∂F∂tX.

Using Schwarz’s formula, the MCF equation and the symmetry of theLevi-Civita connection, one has

∂

∂tgij =

∂

∂t

⟨∂F

∂xi,∂F

∂xj

⟩=

⟨∇t

(∂F

∂xi

),∂F

∂xj

⟩+

⟨∂F

∂xi, ∇t

(∂F

∂xj

)⟩=

=

⟨∇i

(∂F

∂t

),∂F

∂xj

⟩+

⟨∂F

∂xi, ∇j

(∂F

∂t

)⟩=

=

⟨∇i( ~H),

∂F

∂xj

⟩+

⟨∂F

∂xi, ∇j( ~H)

⟩.

As ∂F∂xj

is tangent to M and ~H is normal to M , then⟨~H, ∂F

∂xj

⟩= 0. In

particular,

0 =∂

∂xi

⟨~H,∂F

∂xj

⟩=

⟨∇i( ~H),

∂F

∂xj

⟩+

⟨~H, ∇i

∂F

∂xj

⟩=

=

⟨∇i( ~H),

∂F

∂xj

⟩+

⟨~H, ∇ ∂F

∂xi

∂F

∂xj

⟩.

Therefore ⟨∇i( ~H),

∂F

∂xj

⟩= −

⟨~H, ∇ ∂F

∂xi

∂F

∂xj

⟩.


And so, substituting in the previous equation, we obtain

∂

∂tgij = −

⟨~H, ∇ ∂F

∂xi

∂F

∂xj

⟩−⟨∇ ∂F

∂xj

∂F

∂xi, ~H

⟩=

= −⟨~H,

(∇ ∂F

∂xi

∂F

∂xj

)⊥⟩−⟨(∇ ∂F

∂xj

∂F

∂xi

)⊥, ~H

⟩=

= −⟨~H, II

(∂

∂xi,∂

∂xj

)⟩−⟨II

(∂

∂xj,∂

∂xi

), ~H

⟩=

= −〈 ~H,Aij〉 − 〈Aji, ~H〉 = −2〈 ~H,Aij〉,

where we have used that, by the symmetry of II: Aij = Aji.

Our next step is to study the evolution of√

det g 5; the volumenform:

∂

∂t

√det g =

1

2√

det g

∂

∂tdet g.

In order to compute ∂∂t

det g, we are going to use equation (3.1), Jacobi’sformula for the derivative of a determinant:

∂

∂tdet g = Trace

[adj(g)

∂

∂tg

]= Trace

[(det g)gij · (−2)〈 ~H,Akl〉

]=

= −2 det gTrace

[gij · 〈 ~H,Akl〉

]= −2 det g

n∑i,j=1

gij〈 ~H,Aij〉 =

= −2 det g〈 ~H,n∑

i,j=1

gijAij〉 = −2 det g

⟨~H,

n∑i,j=1

gijII(∂

∂xi,∂

∂xj)

⟩=

= −2 det g

⟨~H,

n∑i,j=1

gij(∇ ∂F

∂xi

∂F

∂xj

)⊥⟩= −2(det g)〈 ~H, ~H〉 =

= −2(det g)| ~H|2,

where we have used the expression of ~H in local coordinates.Therefore,

∂

∂t

√det g =

1

2√

det g

∂

∂tdet g =

1

2√

det g(−2)(det g)| ~H|2 = −| ~H|2

√det g.

5This is a simplified notation, we should write√

det gij(p, t).


3.1. Evolution of the Extrinsic Geometry. Below we will studythe evolution of the extrinsic geometry. For simplicity assume that thecodimension is 1, since in higher codimension many of the quantities in-volved are tensors. The interested reader can find a detailed expositionof the situation in higher codimension in [Smo11].

Let ν ∈ X⊥(M) be a unit normal field. We define the ν-secondfundamental form hν : X(M)× X(M) → C∞(M) as the map given byhν(X, Y ) := 〈ν, II(X, Y )〉.

Notice that hν is a bilinear, symmetric form. The self adjoint oper-ator associated to hν is called the Weiengarten operator and it will bedenoted by Sν . Recall that hν(X, Y ) = 〈Sν(X), Y 〉.

For a compact hypersurface in Rn+1, there is a unique (up to thesign) Gauß map. So, for the sake of simplicity, we will write h and Sinstead of hν and Sν .

Fix p ∈ M and consider a set of normal coordinates around p. Forsimplicity we are going to label Ei = ∂

∂xi. Then we know that Γkij(p) = 0

for all i, j, k (in particular , ∇EiEj(p) = Γkij(p)Ek(p) = 0). It is well

known that if we work with the Levi-Civita connection 6 we can assumethat gij(p) = δij. Finally, let us define

hij := h(Ei, Ej).

At this point we are going to deduce the so called Simons’ identities,that play an important role here and in other key aspects of the theory.These are“Bochner-type” formulae relating the Laplacian of the secondfundamental form and the Hessian of mean curvature.

But first, we are going to get a local expression for the Laplaceoperator in terms of the notation introduced in Remark 3.1. Givenf ∈ C∞(M) and Ei a local basis of smooth tangent fields, then

∆f = Trace(∇2f) = gij∇2ijf = gij(∇Ei∇Ejf −∇∇EiEjf).

Given T is a symmetric 2-tensor on M , then T 2 is the symmetric2-tensor defined (for codimension 1 sub manifolds) by:

T 2(X, Y ) :=∑i

T (X,Ei) · T (Ei, Y ),

6To get the existence of a set of normal coordinates we only need that the con-nection is symmetric, not necessarily torsion free.


where Ei is an arbitrary orthonormal frame. With this notation, wehave that:

Lemma 3.3 (Simons’ Identities). The following identities hold:

(3.3) ∆h = ∇2H +H · h2 − |h|2 · h,

(3.4)1

2∆|h|2 = 〈h,∇2H〉+

∣∣∇h∣∣2 +H Trace(h3)− |h|4,

where 〈h,∇2H〉 =∑

i,j hij∇2ijH

Proof. We are going to work locally around a point p ∈ M , with theorthonormal frame Ei with ∇EiEj(p) = 0 and define

hij := h(Ei, Ej).

Remark 3.4. To avoid confusion, “the squared of hij” will be denotedby (hij)

2, while the coefficients of the tensor h2 will be denoted as h2ij.

We will proceed similarly with h3.

Recall that h2ij = hilg

lmhmj, h3uv = hukg

kihilglmhmv, and |h| =

(gijgklhikhjl)1/2 is the norm of the tensor h with respect to the metric

g.First we prove ∆hij = ∇2

ijH +Hh2ij − |h|2hij.

As we are working in normal coordinates around p,

∆hij = gkl(∇Ek∇Elhij −∇∇EkElhij) = δkl∇Ek∇Elhij = ∇Ek∇Ekhij.

In the abbreviated form we write ∆hij = ∇k∇khij.On the other hand, from the Codazzi equation, we have

(3.5) ∇khij = ∇ihkj.

Thus, we get ∆hij = ∇k∇ihkj.It is not hard to check that ∇k∇ihkj = ∇i∇khkj +Rikjαhαk +Rikkβhβj,where R(Ei, Ek, Ej, Eα) = Rikjα and similarly for R. Taking into ac-count that the ambience space is Euclidean, the Gauß equation gives:

0−Rikjα = 〈Aij, Akα〉 − 〈Aiα, Akj〉,that is,

Rikjα = 〈Aiα, Akj〉 − 〈Aij, Akα〉.By definition, we have Aij = hijν and taking into account that 〈ν, ν〉 =1, then we obtain

Rikjα = 〈Aiα, Akj〉 − 〈Aij, Akα〉 =

= hiαhkj〈ν, ν〉 − hijhkα〈ν, ν〉 = hiαhkj − hijhkα.


Analogously,Rikkβ = hiβhkk − hikhkβ.

Therefore

∆hij = ∇i∇khkj +Rikjαhαk +Rikkβhβj =

= ∇i∇khkj + (hiαhkj − hijhkα)hαk + (hiβhkk − hikhkβ)hβj =

= ∇i∇jhkk + hiαhkjhαk − hijhkαhαk + hiβhkkhβj − hikhkβhβj.In the last equality we have used the symmetry of h and (3.5):

∇khkj = ∇khjk = ∇jhkk.

Recalling that we are using normal coordinates around p, we deduce

H = gijhij = δijhij = hii,

|h|2 = gijgklhikhjl = δijδklhikhjl = hikhik,

h2ij = hilg

lmhmj = hilδlmhmj = hilhlj,

including all this facts in our previous computations, we get

∆hij = ∇i∇jhkk + hiαh2αj − hij|h|2 +Hh2

ij − hikh2kj =

= ∇i∇jH +Hh2ij − |h|2hij =

= ∇2ijH +Hh2

ij − |h|2hij,this concludes the proof of the first identity.

Now we want to prove that

1

2∆|h|2 = hij∇2

ijH +∣∣∇h∣∣2 +H Trace(h3)− |h|4.

As we are working on normal coordinates we have that |h|2 = (hij)2.

Then1

2∆|h|2 =

1

2∆(hij)

2.

Moreover, we already showed that ∆(hij)2 = ∇k∇k(hij)

2, therefore:

1

2∆(hij)

2 =1

2∇k∇k(hij)

2.

Thus,

1

2∆(hij)

2 =1

2∇k∇k(hij)

2 =1

2∇k(2hij∇khij) = (∇khij)

2 + hij∇k∇khij =

= (∇khij)2 + hij(∆hij)

= (∇khij)2 + hij(∇2

ijH +Hh2ij − |h|2hij)

= (∇khij)2 + hij∇2

ijH +Hhijh2ij − |h|4.


where in the last two equalities we have used that:

∆hij = ∇k∇khij = ∇2ijH +Hh2

ij − |h|2hij,as we already checked in the proof of the previous identity.Now, using the definition of trace of h3 and the fact that we are workingwith normal coordinates around p, we get

Trace(h3) = guvh3uv = δuvh3

uv = h3uu = hukg

kihilglmhmu =

= hukδkihilδ

lmhmu = hukhklhlu.

On the other hand,

hijh2ij = hij(hilg

lmhmj) = hij(hilδlmhmj) = hij(hilhlj) = hijhjlhli =

= hukhklhlu,

where we have used that h is symmetric.Therefore, Trace(h3) = hijh

2ij, which implies

1

2∆|h|2 = (∇khij)

2 + hij∇2ijH +H Trace(h3)− |h|4.

Finally, let us check that (∇khij)2 =

∣∣∇h∣∣2. Using our local orthonor-

mal frameEi parallel at p, we have that∣∣∇h∣∣2 =

∑i,j |∇hij|2. Then∣∣∇h∣∣2 =

∑i,j

|∇hij|2

=

⟨∑i,j

〈∇hij, Er〉Er,∑k,l

〈∇hkl, Es〉Es⟩

=

(∑i,j,r

〈∇hij, Er〉)(∑

k,l,s

〈∇hkl, Es〉)〈Er, Es〉

=

(∑i,j,r

〈∇hij, Er〉)(∑

i,j,s

〈∇hij, Es〉)δrs

=∑i,j,r

(〈∇hij, Er〉

)2

=∑i,j,r

(Er(hij)

)2

=∑i,j,r

(∇rhij

)2,

Therefore,

1

2∆|h|2 = hij∇2

ijH +∣∣∇h∣∣2 +H Trace(h3)− |h|4,


as we wanted to prove.

Theorem 3.5 (Evolution of the extrinsic geometry). In our set-ting we have:

(3.6)∂

∂tν = −∂H

∂xi

∂F

∂xjgij = −∇H

(3.7)∂

∂thij = ∆hij − 2Hh2

ij + |h|2hij

(3.8)∂

∂tH = ∆H + |h|2H

(3.9)∂

∂t|h|2 = ∆|h|2 − 2

∣∣∇h∣∣2 + 2|h|4.

Proof. Along this proof, we will denote Ei = ∂F∂xi

.

i) Let us prove that ∂∂tν = − ∂H

∂xi

∂F∂xjgij.

We start by decomposing the vector ∂∂tν in the base Ei,

∂

∂tν = gij

⟨∂

∂tν, Ei

⟩Ej = gij

⟨∇tν, Ei

⟩Ej =

= gij(∂

∂t〈ν, Ei〉 −

⟨ν, ∇tEi

⟩)Ej =

ν ⊥ Ei, implies ∂∂t〈ν, Ei〉 = 0. Moreover, we know that ∇tEi =

∇t∂F∂xi

= ∇i∂F∂t

= ∇i~H = ∇i

~H, which implies

= −gij〈ν, ∇i~H〉Ej =

= −gij(∂

∂xi〈ν, ~H〉 − 〈∇iν, ~H〉

)Ej =

At this point, we use that 〈ν, ~H〉 = 〈ν,Hν〉 = H〈ν, ν〉 = H ·1 = H,

and 〈∇iν, ~H〉 = 〈∇iν,Hν〉 = H〈∇iν, ν〉 = H · 0 = 0 and so:

= −gij ∂H∂xi

∂F

∂xj.

In order to prove that ∂∂tν = −∇H we only have to remember

that, given a smooth function f then ∇f = gij ∂f∂xi

∂∂xj

. Applying


the above expression to H we get:

∂

∂tν = −∂H

∂xi

∂F

∂xjgij = −∇H.

ii) Let us prove now ∂∂thij = ∆hij − 2Hh2

ij + |h|2hij.

∂

∂thij =

∂

∂t

⟨II

(∂

∂xi,∂

∂xj

), ν

⟩=

∂

∂t〈(∇iEj)

⊥, ν〉 =∂

∂t〈∇iEj, ν〉.

By deriving, we get

∂

∂t〈∇iEj, ν〉 = 〈∇t∇iEj, ν〉+ 〈∇iEj, ∇tν〉.

On one hand, we have

∇t∇iEj = ∇i∇tEj = ∇i∇t∂F

∂xj= ∇i∇j

∂F

∂t= ∇i∇j

~H =

= ∇i∇j(Hν).

On the other hand, using (3.6), we deduce

∇tν =∂

∂tν = −grs ∂H

∂xrEs,

which gives

∂

∂t〈∇iEj, ν〉 = 〈∇t∇iEj, ν〉+ 〈∇iEj, ∇tν〉 =

= 〈∇i∇j(Hν), ν〉+ 〈∇iEj,−grs∂H

∂xrEs〉.

Summarizing, we have

∂

∂thij =

∂

∂t〈∇iEj, ν〉 = 〈∇i∇j(Hν), ν〉+ 〈∇iEj,−grs

∂H

∂xrEs〉.

Now, we deal with ∇j(Hν) = (∇jH)ν+H(∇jν). We can compute∇jν as we did in (3.6):

∇jν =∂

∂xjν = glk

⟨∂

∂xjν, El

⟩Ek = glk

(∂

∂xj〈ν, El〉 − 〈ν, ∇jEl〉

)Ek =

= glk(

∂

∂xj0− 〈ν, II

(∂

∂xj,∂

∂xl

)〉)Ek = glk(−hjl)Ek.

Hence, we get

∇j(Hν) = (∇jH)ν +H(∇jν) = (∇jH)ν −HglkhjlEk.


Moreover,

〈∇iEj,−grs∂H

∂xrEs〉 = −grs ∂H

∂xr〈∇iEj, Es〉 = −grs ∂H

∂xr〈(∇iEj)

>, Es〉 =

= −grs ∂H∂xr〈∇iEj, Es〉 = −grs ∂H

∂xr〈ΓkijEk, Es〉 =

= −grs ∂H∂xr

Γkijgks = −Γkijgrsgsk

∂H

∂xr=

= −Γkijδkr

∂H

∂xr= −Γkij

∂H

∂xk.

Therefore,

〈∇i∇j(Hν), ν〉+ 〈∇iEj,−grs∂H

∂xrEs〉 =

= 〈∇i

((∇jH)ν −HglkhjlEk

), ν〉 − Γkij

∂H

∂xk.

At this point, we have that:

∂

∂thij = 〈∇i


), ν〉 − Γkij

∂H

∂xk.

Notice that one has

∇i

((∇jH)ν − (Hglkhjl)Ek

)= ∇i

((∇jH)ν

)− ∇i

((Hglkhjl)Ek

)=

=(∇i(∇jH)

)ν + (∇jH)∇iν −

(∇i(Hg

lkhjl))Ek − (Hglkhjl)∇iEk.

We make the scalar product with ν, taking into account that

〈∇iEk, ν〉 = 〈(∇iEk)⊥, ν〉 =

⟨II

(∂

∂xi,∂

∂xk

), ν

⟩= hik,

and we get

〈∇i


), ν〉 = ∇i(∇jH)−Hglkhjlhik.

In short, we proved that

∂

∂thij = ∇i(∇jH)−Hglkhjlhik − Γkij

∂H

∂xk.

Now observe that by definition of Hessian of a differentiable func-tion we have:∇2f(X, Y ) = X

(Y (f)

)− (∇XY )(f). In particu-

lar, for f = H, X = ∂∂xi

, Y = ∂∂xj

, and taking into account

∇ ∂∂xi

∂∂xj

= Γkij and using the notation ∂f∂xi

= ∇if , one obtains

that

∇i∇jH − Γkij∇kH = ∇2H

(∂

∂xi,∂

∂xj

)= ∇2

ijH.


As g y h are symmetric, glkhjlhik = hikgklhlj = h2

ij.In short,

∂

∂thij = ∇2

ijH −Hh2ij.

Finally, using Simons’ inequality (3.3) one gets

∇2ijH −Hh2

ij = ∆hij − 2Hh2ij + |h|2hij,

which concludes the proof of (3.7).

iii) Now we prove ∂∂tH = ∆H + |h|2H.

This time, we will use that the mean curvature can be also com-puted like H = gijhij.

∂

∂tH =

∂

∂tgijhij =

(∂

∂tgij)hij + gij

(∂

∂thij

).

As we saw in the proof of the previous item,

∂


ij + |h|2hij.

To compute ∂∂tgij recall that we knew, from (3.1), that

∂

∂tgij = −2〈 ~H,Aij〉 = −2〈Hν,Aij〉 = −2H〈ν,Aij〉 = −2Hhij.

As gij = gilglkgkj 7, then

∂

∂tgij =

∂

∂t

(gilglkg

kj

)=

=

(∂

∂tgil)glkg

kj + gil(∂

∂tglk

)gkj + gilglk

(∂

∂tgkj)

=

=

(∂

∂tgil)δjl + gil

(∂

∂tglk

)gkj + δik

(∂

∂tgkj)

=

=∂

∂tgij + gil

(∂

∂tglk

)gkj +

∂

∂tgij =

= 2∂

∂tgij + gil

(∂

∂tglk

)gkj,

so substituting,

∂

∂tgij = −gil

(∂

∂tglk

)gkj = −gil(−2Hhij)g

kj = 2Hgilhijgkj =

= 2Hhikgkj = 2Hhij.

7Indeed, gilglkgkj = δi

kgkj = gij .


Therefore,

∂

∂tH =

(∂

∂tgij)hij + gij

(∂

∂thij

)=

= 2Hhijhij + gij(∆hij − 2Hh2ij + |h|2hij).

Simons’ identity (3.3) implies ∆hij = ∇2ijH +Hh2

ij − |h|2hij, land

so ∆hij − 2Hh2ij + |h|2hij = ∇2

ijH − |h|2hij. Then,

∂

∂tH = 2Hhijhij + gij(∇2

ijH − |h|2hij)

= 2Hhijhij + gij∇2ijH − |h|2gijhij

= 2Hhijhij + Trace(∇2H)− |h|2H= ∆H +H(2hijhij − |h|2).

Note that

|h|2 = gijgklhikhjl = gijglkhkihjl = gjihlihjl = hljhjl = hjlhjl = hijhij,

therefore

2hijhij − |h|2 = 2|h|2 − |h|2 = |h|2.

In short

∂

∂tH = ∆H + |h|2H.

iv) Finally, we will prove ∂∂t|h|2 = ∆|h|2 − 2

∣∣∇|h|∣∣2 + 2|h|4.By definition, we have

|h|2 = gijgklhikhjl.

From the proof of the former item, we already know

∂

∂tgij = 2Hhij,

∂


ij + |h|2hij (it is precisely (3.7)).


Then

∂

∂t|h|2 =

∂

∂t(gijgklhikhjl) =

=

(∂

∂tgij)gklhikhjl + gij

(∂

∂tgkl)hikhjl+

gijgkl(∂

∂thik

)hjl + gijgklhik

(∂

∂thjl

)=

= 2Hhijgklhikhjl + gij2Hhklhikhjl+

gijgkl(∆hik − 2Hh2ik + |h|2hik)hjl+

gijgklhik(∆hjl − 2Hh2jl + |h|2hjl).

If in this last expression we rename the indices of the second termby changing i for k, j for l, k for i and l for j, and in the fourthaddend changing i to j, j for i, k for l and l for k, then you get

∂

∂t|h|2 = 4Hhijgklhikhjl + 2gijgkl(∆hik − 2Hh2

ik + |h|2hik)hjl.

In the proof of Lemma 3.3 we got

h2ij = gklhikhjl,

hijh2ij = Trace(h3),

therefore

4Hhijgklhikhjl = 4H Trace(h3).

Using (3.3),

∆hik − 2Hh2ik + |h|2hik = ∇2

ikH −Hh2ik.

So we get

∂

∂t|h|2 = 4H Trace(h3) + 2gijgkl(∇2

ikH −Hh2ik)hjl.

It is sufficient to check the equality at a point p in which it canbe assumed that are considered normal coordinates (in particular,gij(p) = δij and gkl(p) = δkl), thus the last addend of the aboveexpression can be simplified as follows

2gijgkl(∇2ikH −Hh2

ik)hjl = 2hik(∇2ikH −Hh2

ik) =

= 2hik∇2ikH − 2Hhikh

2ik = 2hik∇2

ikH − 2H Traceh3.

So far, we have proven

∂

∂t|h|2 = 2H Trace(h3) + 2hik∇2

ikH.(3.10)


At this point recall that we want to see

∂

∂t|h|2 = ∆|h|2 − 2

∣∣∇h∣∣2 + 2|h|4.

According to Simons’ identity (3.4),

∆|h|2 = 2hij∇2ijH + 2

∣∣∇h∣∣2 + 2H Trace(h3)− 2|h|4,

thereby substituting the expression we want to show, we get

∂

∂t|h|2 = 2hij∇2

ijH + 2H Trace(h3).(3.11)

Comparing (3.10) and (3.11) we see that both coincide. This com-pletes the proof.

4. A comparison principle for parabolic PDE’s

As we have seen in section 2, understanding and working with themean curvature flow involves a good knowledge about parabolic partialdifferential equations. As it is well known, these equations generallycan not be solved, forcing us to look for results about the qualitativebehavior of the solutions. In this section we prove a theorem in thatdirection which is known as the principle of comparison. Throughoutthis section we will use Lieberman’s book [Lie96].

Let Ω be a domain (open, connected subset) in Rn+1. In this settingwe will write the points of Rn+1 as X = (x, t), where x ∈ Rn.

Definition 4.1 (Parabolic Boundary). We define the parabolic bound-ary of Ω, denoted as ∂PΩ, as the set of points X = (x, t) ∈ ∂Ω (thatis, the topological boundary of Ω) such that for all ε > 0 the paraboliccylinder Q(X, ε) contains points in the complement of Ω; the definitionof parabolic cylinder Q(X0, ε) is

Q(X0, ε) := Y ∈ Rn+1 : |Y −X0| < ε, t < t0,

where X0 = (x0, t0) and |(x, t)| = max|x|Rn ,√|t|.

In the simplest case Ω = D × (0, T ), D a domain in Rn and T > 0,we have that the parabolic boundary of Ω coincides with ∂PΩ = BΩ∪SΩ∪CΩ, where BΩ := Ω×0 ( the “bottom” Ω), SΩ := ∂Ω× (0, T )(the “side” of Ω) and CΩ := ∂(Ω)× 0 (the “corner” of Ω.)


Given u ∈ C2,1(Ω) we define the quasi-linear, second-order operatorP as

(4.1) Pu := −∂u∂t

+ aij(X, u,Du)D2iju+ a(X, u,Du).

We assume that aij(X, z, p) y a(X, z, p) are defined for any (X, z, p) ∈Ω× R× Rn.We say that P is parabolic in a subset S of Ω × R × Rn if the matrixwhose coefficients are aij(X, z, p) is positive definite for any (X, z, p) ∈S. We distinguish two especial cases for S:

• If S = Ω× R× Rn, we say that P is parabolic;• If S = (X, z, p) ∈ U × R × Rn : z = u(X), p = Du(X) for

some function u ∈ C1(U) where U ⊂ Ω, then we will say thatP is parabolic at u.

Lemma 4.2. Let A and B symmetric real matrices of order n, with Apositive definite and B negative semidefinite. Then Trace(A ·B) ≤ 0.

Proof. As A is positive definite, then A has a squared root, that wedenote by A1/2, which is regular and symmetric. By Sylvester’s law ofinertia, as B and A1/2B(A1/2)> = A1/2BA1/2 are congruent symmetricmatrices8, then they have the same number of positive, negative andnull eigenvalues. This means that all the eigenvalues µi of A1/2BA1/2

are non-positive. Then, we have

0 ≥n∑i=1

µi = Trace(A1/2BA1/2) = Trace((A1/2B)A1/2) =

= Trace(A1/2(A1/2B)) = Trace(AB)

Theorem 4.3 (Lieberman, Comparison Principle). Let P be aquasi-linear operator like in (4.1). Suppose aij(X, z, p) dos not dependon z and that there exists a positive increasing function K(L) such thata(X, z, p)−K(L) · z is decreasing in z on Ω× [−L,L]×Rn for L > 0.If u and v are functions in C2,1(Ω \ ∂PΩ) ∩ C(Ω) such P is parabolicat either u or v, Pu ≥ Pv on Ω \ ∂PΩ and u ≤ v on ∂PΩ, then u ≤ von Ω.

Proof. First, we fix L in such a way that [−L,L] contains the range ofu and v , i.e., L := maxsupΩ |u|, supΩ |v|.

8Notice that B = P>A1/2BA1/2P ; where P = (A1/2)−1.


Let us define w := (u − v)eλt en Ω, where λ is a real constant to bedetermine later. Notice that u ≤ v on Ω is equivalent to prove thatw ≤ 0 on Ω. From our assumptions, we know that u ≤ v on ∂PΩ, sowe have to prove that:

w ≤ 0 en Ω \ ∂PΩ.

So would finish if we see that w can not have a positive interior max-imum. We proceed by contradiction: Suppose that X0 = (x0, t0) isan interior maximum such that w(X0) > 0. Classical Analysis says tous that if a function of class C2 reaches the maximum at an interiorpoint, then the gradient at that point is zero and the Hessian matrixis negative semidefinite 9. In particular, we have:

(4.2) Dw(X0) = 0⇔ (Du−Dv)(X0)eλt0 = 0⇒ Du(X0) = Dv(X0),

∂w

∂t(X0) = 0⇔ ∂

∂t(u− v)(X0)eλt0 + λ(u− v)(X0)eλt0 = 0,

which implies:

(4.3) λ(u− v)(X0) = − ∂

∂t(u− v)(X0),

(4.4) The matrix (Dijw(X0))i,j=1,...,n is negative semidefinite10.

Below, for convenience, we denote R := (X0, u(X0), Du(X0)) and S :=(X0, v(X0), Dv(X0)). From our hypothesis we know Pu ≥ Pv on Ω \∂PΩ; in particular

0 ≤ Pu(X0)− Pv(X0) =

using the definition of the operator P

= aij(R)Diju(X0)− aij(S)Dijv(X0) + a(R)− a(S)− ∂

∂t(u− v)(X0) =

Notice that the first two terms aij(R) = aij(X0, u(X0), Du(X0)) andaij(S) = aij(X0, v(X0), Dv(X0)) coincide. This is due to the fact that

9We are assuming that n ≥ 2. If n = 1 would have that the derivative vanishesat that point and the second derivative is positive at that point. We no longer dothis distinction, but it is clear that what follows is valid in the case n = 1 with theobvious changes.

10IfX0 is an interior maximum of w, then the Hessian matrix w inX0, Hw(X0) =(Dijw(X0))i,j=1,...,n,t, is negative semidefinite. Therefore, by the criterion of theprincipal minors, the square submatrix obtained by removing the last row and lastcolumn of Hw(X0), which is (Dijw(X0))i,j=1,...,n, remains a negative semidefinitematrix.


aij does not depend on the second argument and because, by(4.2),Du(X0) = Dv(X0). Using the above fact and (4.3) we have:

= aij(R)Dij(u− v)(X0) + a(R)− a(S) + λ(u− v)(X0) ≤

Applying Lemma 4.2 to the matrices A := (aij(R))i,j=1,...,n and B :=(Dji(u− v)(X0))i,j=1,...,n, we get aij(R)Dij(u− v)(X0) ≤ 0, and so

≤ a(R)− a(S) + λ(u− v)(X0) ≤

At this point, recall that w(X0) > 0 , i.e., u(X0) > v(X0). Asa(x, z, p) − K(L)z is a decreasing function of z in Ω × [−L,L] × Rn,then a(R) − K(L)u(X0) ≤ a(S) − K(L)v(X0), or in other words,a(R)− a(S) ≤ K(L)(u− v)(X0),

≤ K(L)(u− v)(X0) + λ(u− v)(X0) ≤ [K(L) + λ](u− v)(X0).

Now, It suffices to take λ < −K(L) to get that (u− v)(X0) ≤ 0, thatis, w(X0) ≤ 0, which is contrary to w(X0) > 0. This contradictionproves the theorem.

5. Graphical submanifolds. Comparison Principle andConsequences

The mean curvature flow has been extensively studied in some familieswhich have specific geometric conditions, as is the case of hypersurfaces,Lagrangian submanifolds, graphs, etc. In this section we will focus onthe study of smooth graphs.

Let u : Rn → R be a smooth function. It is well known that the graphof u, Graph(u) = (x, u(x)) : x ∈ Rn is a hypersurface Rn+1. Then,we can study how it evolves under mean curvature flow. The first goalof this section will be to deduce the evolution equation for ut. Next,we will see that we can apply a comparison principle to obtain a resultknown as avoidance principle.

We start with a graph M0 = (x, u(x)) : x ∈ Rn ⊂ Rn+1. We arelooking for a map

F : Rn × [0, T )→ Rn+1

F (p, t) = (x(p, t), u(x(p, t), t))

satisfying the MCF equation

∂F

∂t= Hν,


and such that F (·, 0) = M0 = Graph(u). At this point, it would be ex-tremely useful to know how to express the main geometrical quantitiesassociated to M0 in terms of u and its derivatives. This is the purposeof the next lemma

Lemma 5.1 (Graphical submanifolds). Let u : Rn → R be a smoothfunction and M0 = Graph(u). Then, we have:

(1) gij = δij +DiuDju,

(2) gij = δij −DiuDju

1 + |Du|2,

(3) ν =(−Du, 1)√1 + |Du|2

,

(4) hij =D2iju√

1 + |Du|2,

(5) H = div

(Du√

1 + |Du|2

).

Proof.

(1) The proof of the first item is straightforward:

gij = 〈DiF,DjF 〉 = 〈(ei, Diu), (ej, Dju)〉 = δij +DiuDju,

where e1, . . . en+1 denotes the canonical basis of Rn+1.(2) If we make the matrix product, we get

gikgkj = (δik +DiuDku)(δkj −

DkuDju

1 + |Du|2) =

= δikδkj − δikDkuDju

1 + |Du|2+DiuDkuδkj −DiuDku

DkuDju

1 + |Du|2=

= δij −DiuDju

1 + |Du|2+DiuDju− |Du|2

DiuDju

1 + |Du|2=

= δij − (1 + |Du|2)DiuDju

1 + |Du|2+DiuDju =

= δij −DiuDju+DiuDju = δij,

which means that (gij) is the inverse matrix of (gij).(3) The vectors DiF = (ei, Diu), i = 1, . . . n are a global basis

of the tangent bundle of Graph(u).Then (−Du, 1) is a normal,


non-vanishing vector field. So, we only have to normalized it

ν =(−Du, 1)√1 + |Du|2

.

(4)

hij = 〈II(DiF,DjF ), ν〉 = 〈(∇DiFDjF )⊥, ν〉 = 〈∇DiFDjF, ν〉 =

= 〈DDiFDjF, ν〉 = 〈D2ijF, ν〉 =

⟨(0, D2

iju),(−Du, 1)√1 + |Du|2

⟩=

=D2iju√

1 + |Du|2.

(5) On one hand, we have

div

(Du√

1 + |Du|2

)= Di

(Diu√

1 + |Du|2

)=

=

∑ni=1 D

2iiu√

1 + |Du|2− 1

2Diu

1

(1 + |Du|2)3/22〈Di(Du), Du〉 =

=∆u√

1 + |Du|2−Diu

1

(1 + |Du|2)3/2〈D2

jiuDju,Du〉 =

=∆u√

1 + |Du|2−

DiuD2jiuDju

(1 + |Du|2)3/2=

∆u√1 + |Du|2

−DiuDjuD

2iju

(1 + |Du|2)3/2.

We number this auxiliary result that we will use it a few timesin later calculations:

(5.1) div

(Du√

1 + |Du|2

)=

∆u√1 + |Du|2

−DiuDjuD

2iju

(1 + |Du|2)3/2.

On the other hand,

H = gijhij =

(δij −

DiuDju

1 + |Du|2

)D2iju√

1 + |Du|2=

= δijD2iju√

1 + |Du|2−

DiuDjuD2iju

(1 + |Du|2)3/2=

∆u√1 + |Du|2

−DiuDjuD

2iju

(1 + |Du|2)3/2.

Therefore,

H = div

(Du√

1 + |Du|2

).


Now we use this information to find new partial differential equationsfor graphs that evolve by mean curvature. We start by deriving withrespect to time the application F . Deriving component by componentand applying the chain rule,

∂F

∂t=

(∂x

∂t,∂u

∂xi

∂xi∂t

+∂u

∂t

)=

(∂x

∂t,

⟨Du,

∂x

∂t

⟩+∂u

∂t

).

Using this, the equation of the mean curvature flow becomes:(∂x

∂t,

⟨Du,

∂x

∂t

⟩+∂u

∂t

)=

H√1 + |Du|2

(−Du, 1).

or equivalently

(5.2)

(5.3)

∂x

∂t= −H Du√

1 + |Du|2,⟨

Du,∂x

∂t

⟩+∂u

∂t=

H√1 + |Du|2

.

Notice that, using 5.2, one has⟨Du,

∂x

∂t

⟩= − H√

1 + |Du|2〈Du,Du〉 = − H√

1 + |Du|2|Du|2.

Therefore, equation (5.3) can be written as follows:

(5.4)∂u

∂t= H(1 + |Du|2)

1√1 + |Du|2

= H√

1 + |Du|2

Finally, substituting H = div

(Du√

1+|Du|2

)in (5.2) and (5.4) they be-

come:

(5.5)

(5.6)

∂x

∂t= − div

(Du√

1 + |Du|2

)· Du√

1 + |Du|2,

∂u

∂t= div

(Du√

1 + |Du|2

)·√

1 + |Du|2.

We need some extra computations to get an expression for the abovedivergence. According to (5.1),

div

(Du√

1 + |Du|2

)=

∑ni=1 D

2iiu√

1 + |Du|2−

DiuDjuD2iju

(1 + |Du|2)3/2.


Then,

div

(Du√

1 + |Du|2

)=

∑ni=1D

2iiu√

1 + |Du|2−

DiuDjuD2iju

(1 + |Du|2)3/2=

=1√

1 + |Du|2

(δijD

2iju−

DiuD2jiuDju

1 + |Du|2

)=

=1√

1 + |Du|2

(δij −

DiuDju

1 + |Du|2

)D2iju.

Substituting in (5.6),

(5.7)∂u

∂t=

(δij −

DiuDju

1 + |Du|2

)D2iju = ∆u− D2u(Du,Du)

1 + |Du|2,

where recall that D2u means the Hessian operator associated to u.

With all we have seen so far we can prove the following well-knownresult for mean curvature flow of a sphere.

Proposition 5.2. The spheres of Euclidean space evolve under themean curvature flow as spheres that concentrically contract until col-lapse in finite time at one point; the common center of the family ofspheres.

Proof. The upper n-dimensional hemisphere of radius ρ can be seen asthe graph of the function u : B(0, ρ)→ R given by u(x) :=

√ρ2 − |x|2,

where B(0, ρ) ⊂ Rn means the Euclidean ball centered at the origin ofradius ρ.The abundance of symmetry of the sphere will allow us to solve thepartial differential equation of the mean curvature flow in this particularcase. Indeed,

Diu = − xi√ρ2 − |x|2

,

and from here we get

Du = − x√ρ2 − |x|2

, |Du|2 =|x|2

ρ2 − |x|2, 1 + |Du|2 =

ρ2

ρ2 − |x|2;

gij = δij +xixj

ρ2 − |x|2;

gij = δij −(ρ2 − |x|2

ρ2

xixjρ2 − |x|2

)= δij −

xixjρ2

;


ν =

√ρ2 − |x|2ρ

(− −x√

ρ2 − |x|2, 1

)=

(x

ρ,

√ρ2 − |x|2ρ

).

Thus

D2iju =

∂

∂xj

(∂

∂xiu

)=

∂

∂xj

(− xi√

ρ2 − |x|2

)=

= −δij√ρ2 − |x|2 − xi

(− xj√

ρ2−|x|2

)ρ2 − |x|2

=

= − δij√ρ2 − |x|2

− xixj(ρ2 − |x|2)3/2

,

therefore

hij = −√ρ2 − |x|2ρ

(δij√

ρ2 − |x|2+

xixj(ρ2 − |x|2)3/2

)= −1

ρ

(δij+

xixjρ2 − |x|2

).

Using that H = gijhij, we get:

H = gijhij = −∑i,j

(δij −

xixjρ2

)1

ρ

(δij +

xixjρ2 − |x|2

)=

= −1

ρ

(δijδij + δij

xixjρ2 − |x|2

− xixjρ2

δij −∑

i,j x2ix

2j

ρ2(ρ2 − |x|2)

)=

= −1

ρ

(∑i,j

δij +|x|2

ρ2 − |x|2− |x|

2

ρ2− |x|4

ρ2(ρ2 − |x|2)

)=

= −1

ρ

(n+ |x|2ρ

2 − ρ2 + |x|2

ρ2(ρ2 − |x|2)− |x|4

ρ2(ρ2 − |x|2)

)=

= −nρ,

where we have used δijδij =∑

i,j δij, δijxixj = |x|2, and∑

i,j x2ix

2j =∑

i x2i

∑j x

2j = |x|2|x|2 = |x|4.

Taking into account the previous computations the PDE (5.4) leadsto a partial differential equation that we solve (notice that we do not

work now with u(x) but with u(x, t) =√(

ρ(t))2 −

∣∣x(t)∣∣2) :

∂u

∂t= H

√1 + |Du|2 = − n

ρ(t)

ρ(t)√(ρ(t)

)2 −∣∣x(t)

∣∣2 = − n√(ρ(t)

)2 −∣∣x(t)

∣∣2 ,


in other words,∂u

∂t= − n

u(x, t),

which is an ODE that we can integrate:

udu = −ndt⇒ u2

2= −nt+ C ⇒ u(x, t) =

√K(x)− 2nt,

where K(x) is a “constant” (which depends on x but not on t). As the

initial data is u(x, 0) =√ρ2 − |x|2, then we deduce

u(x, t) =√(

ρ2 − 2nt)− |x|2, x ∈ B(0, ρ2 − 2nt).

Finally, notice that:

(5.8) ρ2 − 2nt ≥ 0⇔ ρ2

2n≥ t,

then, if the starting sphere is Sρ(0), the collapse occurs at time t = ρ2

2n(see Figure 2.)

Let us turn our attention to equation (5.7):

∂u

∂t=

(δij −

DiuDju

1 + |Du|2

)D2iju.

If we compare it with the definition of the operator (4.1) we obtainthat (in this particular case):

aij(X, z, p) = δij −pipj

1 + |p|2(therefore it does not depend on z),

a(X, z, p) ≡ 0.

We claim that (5.7) defines a parabolic operator, that is, that the ma-trix whose coefficients are aij(X, z, p) is positive definite. Indeed,

• At p = 0, aij = δij, which is the identity matrix;• If p 6= 0, consider x ∈ Rn \ 0.

xiaij(X, z, p)xj = xi

(δij −

pipj1 + |p|2

)xj = xiδijxj −

xipipjxj1 + |p|2

=

= 〈x, x〉Rn −(xipi)(pjxj)

1 + |p|2= 〈x, x〉Rn −

〈x, p〉Rn〈x, p〉Rn1 + |p|2

=

= |x|2 − 〈x, p〉2

1 + |p|2.


Figure 2. A family of concentric spheres collapsing atone point.

Taking this into account,

xiaij(X, z, p)xj > 0⇔ |x|2 − 〈x, p〉

2

1 + |p|2> 0⇔ 〈x, p〉2 < |x|2(1 + |p|2),

and this last inequality holds by the Cauchy-Schwarz inequality(in Rn): 〈x, p〉2 ≤ 〈x, x〉〈p, p〉2 = |x|2|p|2, and because, obvi-ously, |x|2|p|2 < |x|2(1 + |p|2) (recall that |x| 6= 0).

At this point we are ready to prove the comparison principle.

Theorem 5.3 (Comparison principle). Let M0 and N0 be compact,embedded hypersurfaces, without boundary, in Rn+1 that do not inter-sect. If Mt and Nt are their respective evolutions by the mean curvatureflow, then they never intersect.


First proof. We proceed by contradiction. Assume Mt and Nt first in-tersect at time t0 at a point p (which is an interior point of both surfacesbecause , by assumption, none of them have boundary.) Then both hy-persurfaces have the same tangent plane at the point p, otherwise thiswould not be your first point of contact. Then Mt0 y Nt0 can be ex-pressed locally about p as graphs of functions, ut0 and vt0 respectively.As both hypersurfaces have the same tangent plane at p, also have thesame unit normal vector at p, except perhaps by the sign. Consideringthe same orientation on Mt0 and Nt0 we can compare them. Note that,as t0 is the first point of contact, just a moment before both hypersur-faces do not intersect, , so either vt < ut or vt > ut. Without loss ofgenerality (since this depends on the choice of the Gauß map) we canassume vt > ut. So, just a moment before t0, there exists ε > 0 suchthat vt − ut > ε.

Now we apply the mean curvature flow starting in that instant be-fore t0. As we have seen before, we know that ut and vt verifies thequasilinear parabolic equation:

P u = −∂u∂t

+

(δij −

Diu,Dju

1 + |Du|2

)D2iju = 0.

As we also check before, P can be seen as an operator satisfying the hy-potheses of the comparison principle for parabolic operators (Theorem4.3.) It is obvious that vt − ε is also a solution of the above equation.Moreover, ut y vt−ε satisfy the assumptions of Theorem 4.3 in a neigh-borhood of the point p, then we deduce that ut ≤ vt − ε(< vt) in thisparticular neighbourhood of p, which contradicts u(p, t0) = v(p, t0).This contradiction completes the proof.

As the reader can see, the idea in the proof of the above theorem isvery simple. However, we may get a better result if we work a littlemore on the techniques to compare solutions of a partial differentialequation of parabolic type. For this purpose we are going to followMantegazza’s monograph [Man11]. But first recall the concept of lo-cally Lipschitz function, then we will use it in a previous result knownas Hamilton’s Trick.

Definition 5.4. Let (A, d) be a metric space. A function f : A→ R isdefined to be Lipschitz (globally on A) if there exists a constant L > 0such that:

|f(x)− f(y)| ≤ L · d(x, y) ∀x, y ∈ A.


We will say that f is locally Lipschitz if for each x0 ∈ A there existU0 a neighborhood of x0 and a constant L0 > 0 such that:

|f(x)− f(y)| ≤ L0 · d(x, y) ∀x, y ∈ U0.

Remark 5.5.

(1) If we want to be more precise, then we say that f is (locally)L-Lipschitz, specifying the Lipschitz constant.

(2) It can be shown that a Lipschitz function f : R→ R is differen-tiable almost everywhere (eg, proving that a Lipschitz functionis absolutely continuous). Later we will use this fact.

Lemma 5.6 (Hamilton’s Trick). Let u : M × (0, T ) → R a C1-function such that for each t0 there exist δ > 0 and a compact subsetK ⊂ M \ ∂M such that for any t ∈ (t0 − δ, t0 + δ) the maximumumax(t) := maxp∈M u(p, t) is reached at least at one point of K.Then, the function umax is locally Lipschitz in (0, T ) and for each t0 ∈(0, T ) where it is differentiable we have:

dumax(t0)

dt=∂u(p, t0)

∂t

where p ∈M \ ∂M is any interior point where u(·, t0) reaches its max-imum.

Proof. Consider t0 ∈ (0, T ). Let δ and K be as in the hypotheses ofthe lemma.We start by showing that u|K×(t0−δ,t0+δ) is Lipschitz with respect to t.Fix p ∈ K, then we have to deduce the existence of a constant C > 0such that if t1 < t2 in (t0 − δ, t0 + δ) then

|u(p, t2)− u(p, t1)| ≤ C|t2 − t1|.This is essentially a consequence of the Mean Value Theorem for func-tions of class C1. Indeed, without loss of generality we can assumethat [t0 − δ, t0 + δ] ⊂ (0, T ). As u is C1, we can apply the Mean ValueTheorem to the function u(p, ·) : [t0 − δ, t0 + δ] → R. Furthermore∂u∂t

: K × [t0 − δ, t0 + δ] → R is bounded (K is compact). This im-

plies the existence of a constant C > 0 such that∣∣∂u(p,t)

∂t

∣∣ ≤ C for all(p, t) ∈ K × [t0 − δ, t0 + δ]. Therefore,

|u(p, t2)− u(p, t1)| =∣∣∣∣u(p, t2)− u(p, t1)

t2 − t1

∣∣∣∣|t2 − t1| = ∣∣∣∣∂u(p, t3)

∂t

∣∣∣∣|t2 − t1|≤ C|t2 − t1|,


as we wanted to prove.

Let us see that umax is locally Lipschitz in (0, T ). Take t0 in (0, T )joint with δ and K provided by the hypotheses of the lemma. Consider0 < ε < δ. Taking into account that u|K×(t0−δ,t0+δ)is Lipschitz withrespect to t, we have

umax(t0 + ε) = u(q, t0 + ε) ≤ u(q, t0) + εC ≤ umax(t0) + εC,

for some q ∈ K (this point q exists by hypothesis). So

umax(t0 + ε)− umax(t0)

ε≤ C.

Analogously,

umax(t0) = u(p, t0) ≤ u(p, t0 + ε) + εC ≤ umax(t0 + ε) + εC,

for a certain p ∈ K. Therefore

umax(t0)− umax(t0 + ε)

ε≤ C.

Summarizing, we have showed that 0 < ε < δ,

|umax(t0)− umax(t0 + ε)| ≤ C|(t0 + ε)− t0|.If we consider −δ < ε < 0, then we can prove in a similar way that:

|umax(t0)− umax(t0 + ε)| ≤ C|(t0 + ε)− t0|.Thus, we have got that umax is locally Lipschitz in (0, T ), which impliesthat it is differentiable a.e. in t ∈ (0, T ).

Finally, take a point t0 ∈ (0, T ) where umax is differentiable. Fromour assumptions, there exists p ∈M \ ∂M so that umax(t0) = u(p, t0).By the Mean Value Theorem, for each 0 < ε < δ there exists ξ ∈(t0, t0 + ε) such that u(p, t0 + ε) = u(p, t0) + ε∂u(p,ξ)

∂t. Therefore

umax(t0 + ε) ≥ u(p, t0 + ε) = u(p, t0) + ε∂u(p, ξ)

∂t= umax(t0) + ε

∂u(p, ξ)

∂t,

from which, as ε > 0, we can deduce


ε≥ ∂u(p, ξ)

∂t.

Taking limit, as ε→ 0 we get

u′max(t0) ≥ ∂u(p, ξ)

∂t.

Applying just the same argument for −δ < ε < 0 we conclude


ε≤ ∂u(p, ξ)

∂t,


and taking limit again, as ε→ 0 what we get now is that:

u′max(t0) ≤ ∂u(p, ξ)

∂t.

Summarizing, u′max(t0) = ∂u(p,ξ)∂t

.

Corollary 5.7. Hamilton’s trick also holds if we consider umin(t) :=minp∈M u(p, t) instead of umax.

Proof. We just consider v := −u.

Theorem 5.8 (Comparison principle). Let ϕ : M1× [0, T )→ Rn+1

and ψ : M2 × [0, T ) → Rn+1 be two hypersurfaces moving by meancurvature. Suppose that M1 is compact, M2 is complete and that ψt isproper11 for any t ∈ [0, T ).Then the distance between the hypersurfacesis non-decreasing in time.

Proof. First notice that, as ϕt(M1) is compact and ψt(M2) is properlyimmersed, then the distance between both hypersurfaces at time t isgiven by:

d(t) = minp∈M1,q∈M2

|ϕ(p, t)− ψ(q, t)|.

We want to apply Lemma 5.6, but the problem is that the Euclideannorm has differentiability problems at the origin. So we are going toconsider M := M1 ×M2 and the function u : M × (0, T ) → R givenby u(p, q, t) := |ϕ(p, t) − ψ(q, t)|2 = 〈ϕ(p, t) − ψ(q, t), ϕ(p, t) − ψ(q, t)〉Notice that

umin(t) := min(p,q)∈M

|ϕ(p, t)− ψ(q, t)|2 =(d(t))2.

So, it suffices to prove that umin(t) is a non-decreasing function.

Step 1. The function u satisfies the hypotheses of Corollary 5.7 (whichare the same as Lemma 5.6.)

Clearly u is C1 (even more, it is smooth.)

Fix t0, we want to obtain the existence of δ > 0 and a compactsubset K ⊂M \ ∂M such that for any t ∈ (t0− δ, t0 + δ) the minimumumin(t) = min(p,q)∈M u(p, q, t) is attained at one point of K.

11ψt : M2 → Rn+1 is defined proper if for any compact subset K ⊂ Rn+1 thenψ−1

t (K) is also compact.


Take δ > 0 satisfying [t0 − δ, t0 + δ] ⊂ (0, T ). Notice that umin([t0 −δ, t0 + δ]) is bounded, because M1 is compact. So, there is a positiveconstant β > 0 such that

umin([t0 − δ, t0 + δ]) ⊂ [0, β].

Let us define:

K := u−1([0, β])

)∩ (M1 ×M2 × [t0 − δ, t0 + δ]) .

We have to prove that K is compact. Clearly, K is closed. If weprove that K is bounded, then we have done. Assume that K is notbounded. As M1 and [t0−δ, t0 +δ] are compact, this means that π2(K)is unbounded, where π2 is the second canonical projection. Thus, wetake qn a sequence in π2(K) such that neither the sequence itself norany subsequence is bounded. Consider pn ∈M1 and tn ∈ [t0− δ, t0 + δ]such that (pn, qn, tn) ∈ K. Up to taking a subsequence, we can assumethat tn → t′ ∈ [t0 − δ, t0 + δ]. From the definiton of K, we have thatψtn(qn) belongs to the set

K :=x ∈ Rn+1 : distRn+1 (x, ϕ (M1 × [t0 − δ, t0 + δ])) ≤

√β,

which is compact. As we are assuming that ψt′ is proper, then K′ =ψ−1t′ (K) is also compact. So, any limit point of qnmust lie on K′, which

is absurd because we are assuming that this sequence is unbounded.This contradiction proves that K is compact.

Hence, Corollary 5.7 gives us the function umin is locally Lipschitz in(0, T ) and for each t0 ∈ (0, T ) where it is differentiable we have:

d

dtumin(t0) =

∂

∂tu(p0, q0, t0),

where (p0, q0) ∈M1×M2 is any point where u(·, t) reaches its minimum.

Step 2. Let (p0, q0) be a minimum of u(·, t), then∂

∂tu(p0, q0, t0) ≥ 0.

We distinguish two cases:

Case 1. u(p0, q0, t0) = 0, then ∂∂tu(p0, q0, t0) cannot be negative,

otherwise u(p0, q0, t) would be negative for t in a small interval [t0, t0 +s), which is absurd.

Case 2. u(p0, q0, t0) 6= 0. As (p0, q0) is a minimum of u, we havethat ϕ(p0, t0)− ψ(q0, t0) is normal to both hypersurfaces; ϕt0(M1) andψt0(M2). In other words, the respective tangent hyperplanes are paral-lel.


This allows us to write the respective hypersurfaces (locally aroundp0 and q0) as graphs of functions, f and h, defined on (a part of) one oftheir tangent spaces, Π, in a small time interval (t0−ε, t0 +ε). Withoutloss of generality, we can fix a reference in Rn+1 so that e1, . . . , en,the canonical basis of Rn, is a base of the hyperplane Π. Assume thatϕ(p0, t0) = (0, f(0, t0)) and ψ(q0, t0)) = (0, h(0, t0)) with f(0, t0) >h(0, t0). Note that in this reference we have

en+1 =ϕ(p0, t0)− ψ(q0, t0)

|ϕ(p0, t0)− ψ(q0, t0))|.

From (5.7) we have:

(5.9)∂f

∂t= ∆f − ∇

2f(Df,Df)

1 + |Df |2,

∂h

∂t= ∆h− ∇

2h(Dh,Dh)

1 + |Dh|2.

On the other hand, as the function f(x, t0) − h(x, t0) has a minimumat x = 0, then its gradient vanishes at this point and the Hessian ispositive semidefinite at x = 0. In particular, the Laplacian of f(x, t0)−h(x, t0) is non-negative.

0 = ∇(f(0, t0)− h(0, t0)

)= ∇f(0, t0)−∇h(0, t0),

0 ≤ ∆(f(0, t0)− h(0, t0)

)= ∆f(0, t0)−∆h(0, t0).

Moreover, from our choice of the set of coordinates, we also have;

∇f(0, t0) = ∇h(0, t0) = 0.

Using (5.1), and Df(0, t0) = ∇f(0, t0) = 0, we have

H(0, t0) = div

(Df

1 + |Df |2

)(0, t0) =

=

(∆f√

1 + |Df |2−

DifD2ijfDjf

(1 + |Df |2)3/2

)(0, t0) = ∆f(0, t0).

Again, all we got for f is also valid for h.If we denote as νϕ and νψ the unit normal fields associated to the ϕand ψ, respectively, we can write

Hϕ(p0, t0)νϕ(p0, t0) =(0,∆f(0, t0)

),

Hψ(q0, t0)νψ(q0, t0) =(0,∆h(0, t0)

),

If we multiply by en+1, then we obtain the following equalities:

∆f(0, t0) = Hϕ(p0, t0)〈νϕ(p0, t0), en+1〉,

∆h(0, t0) = Hψ(q0, t0)〈νψ(q0, t0), en+1〉.


Hence,

(5.10) 〈Hϕ(p0, t0)νϕ(p0, t0)−Hψ(q0, t0)νψ(q0, t0), en+1〉 =

= ∆f(0, t0)−∆h(0, t0) ≥ 0.

Now, we can compute ∂∂tu(p0, q0, t0), taking into account that ϕ and

ψ are solutions of the MCF equation:

∂

∂tu(p0, q0, t) =

∂

∂t〈ϕ(p0, t)− ψ(q0, t), ϕ(p0, t)− ψ(q0, t)〉 =

= 2

⟨∂ϕ(p0, t)

∂t− ∂ψ(q0, t)

∂t, ϕ(p0, t)− ψ(q0, t)

⟩=

= 2〈Hϕ(p0, t)νϕ(p0, t)−Hψ(q0, t)ν

ψ(q0, t), ϕ(p0, t)− ψ(q0, t)〉 =

= 2⟨Hϕ(p0, t)ν

ϕ(p0, t)−Hψ(q0, t)νψ(q0, t), en+1

⟩|ϕ(p0, t0)− ψ(q0, t0)|,

where we have used that en+1 = ϕ(p0,t0)−ψ(q0,t0)|ϕ(p0,t0)−ψ(q0,t0))| .

Evaluating the last equality at t = t0 and taking (5.10) into account,we obtain that ∂

∂tu(p0, q0, t0) ≥ 0.

As we noted at the beginning of the proof, the second step provesthat d(t) is non-decreasing, which completes the demonstration.

Remark 5.9 ([MSHS14]). We would like to point out that the proper-ness assumption cannot be relaxed with that of completeness. Indeed,take as f : M1 → R3 be the unit euclidean sphere and as g : M2 → R3

a complete minimal surface lying inside the unit ball. Such exampleswere first constructed by Nadirashvili [Nad96]. Obviously f and g donot have intersection points. However, under the mean curvature flow,f shrinks to a point in finite time while g remains stationary.

The following result is an immediate consequence of the previous the-orem, but it is useful to state it independently. It has a very geometricmeaning and it will be used in several arguments.

Corollary 5.10 (Inclusion principle). Let ϕ : M1 × [0, T ) → Rn+1

and ψ : M2× [0, T )→ Rn+1 two compact hypersurfaces moving by meancurvature. Assume that the inner domain of ϕ(M1, 0) strictly containsψ(M2, 0). Then, ψ(M2, t) stays strictly inside of ϕ(M1, t) for any t ∈[0, T ).

Among other things, the above corollary has an interesting conse-quence, which has a decisive influence on the study of mean curvature


flow; the existence of singularities in finite time for the flow of a com-pact hypersurface.

Corollary 5.11 (Existence of singularities in finite time). Let Mbe a compact hypersurface in Rn+1. If Mt represents its evolution bythe mean curvature flow, then Mt must develop singularities in finitetime. Moreover, if we denote this maximal time as Tmax, then we havethat:

2nTmax ≤ (diamRn+1(M))2 .

Proof. As M in compact, then it can be included inside an open ballB(p, ρ). So, M must develop a singularity before the flow of Snp collapsesat the point p, otherwise we would contradict Corollary 5.10. The upperbound of Tmax is just a consequence of (5.8).

A natural question is: What can we say when M is not compact?In this case, we can have long time existence. A trivial example isthe case of a complete, properly embedded minimal hypersurface Min Rn+1. Under the mean curvature flow, M remains stationary, sothe flow exists for any value of t. If we are looking for non-stationaryexamples, then we can consider the following example:

Example 5.12 (Grim hyperplanes). Consider the euclidean prod-uct M = Γ × Rn−1, where Γ is the grim reaper in R2 represented bythe immersion f : (−π/2, π/2)→ R2 given by

f(x) = (x, 1− log cosx).

If we move M by mean curvature we get Mt = φt(M) + t · en+1, whereagain e1, . . . , en+1 represents the canonical basis of Rn+1 and φt :M → M is a (tangent) diffeomorphism. In other words, M moves byvertical translations, that do not have singularities for any value of t.By definition, we say that M is a translating soliton in the directionof en+1. More generally, any translator in the direction of en+1 whichis a Riemannian product of a planar curve and an euclidean spaceRn−1 can be obtained from this example by a suitable combination ofa rotation and a dilation (see [MSHS14] for further details.) Each ofthese translators will be called a grim hyperplane.

The ideas introduced in the proof of Theorem 5.8 can be used to geta very interesting result.

Theorem 5.13 (Embeddedness principle). Let F : M× [0, T )→ Rn+1

a MCF, with M compact. If F0 : M → Rn+1 is an embedding, then Ftis an embedding for any t ∈ [0, T ).


Figure 3. A grim hyperplane

Proof. Let us define

A := t ∈ [0, T ) : Fs is an embedding for all s ∈ [0, t].

Following a classical connectedness argument, we are going to provethat A is open, closed and non-empty, so A = [0, T ).

Notice that A is not empty, since 0 ∈ A .

Moreover A is open. Indeed, take t ∈ A and assume his set is notopen. That would mean the existence of a sequence tjj∈N t andsequences of points pjj∈N and qjj∈N in M such that F (pj, tj) =F (qj, tj), pj 6= qj, for all j ∈ N. As M is compact, then (up to asubsequence) we can assume that pjj∈N → p and qjj∈N → q. Wehave two possibilities; either p = q or p 6= q.

If p 6= q, then we would conclude that Ft is not an embedding whichid contrary to t ∈ A .

If p = q, we can take U a neighborhood of p in M such that Ftj |U isan embedding for all j ∈ N. This is possible since Ftj are immersionsand they converge to Ft which is an embedding. If j is large enough,then we have that pj and qj belong to U , which is absurd becauseF (pj, tj) = F (qj, tj). This contradiction proves the openness of A .


Note that so far we have not used that F : M × [0, T )→ Rn+1 movesby its mean curvature.

Finally, we are going to prove that A is closed, or equivalently thatsup A = T. We proceed again by contradiction. Suppose t0 = sup A <T.

So, we consider W ⊂M ×M ,

W := (p, q) ∈M ×M : F (p, t0) = F (q, t0), p 6= q.

Then, W is a closed subset disjoint from the diagonal ∆ = (p, p) :p ∈ M. Indeed, if (p, q) is a point in W , we only have to check thatp 6= q to guarantee that (p, q) ∈ W (the other condition is closed.) Let(p,qn) ⊂ W such that (p,qn) → (p, q). If p = q, then take a chart(V, φ) around p in M . There exists n0 ∈ N such that, for any n ≥ n0,one has pn, qn ∈ V . Up to a subsequence, we have that

vn :=φ(pn)− φ(qn)

|φ(pn)− φ(qn)|

converges to some unitary vector v ∈ Sn.. Then

d(Ft0 φ−1

)φ(p)

(v) = limn→∞

F (pn, t0)− F (qn, t0)

|φ(pn)− φ(qn)|= 0,

which is contrary to the assumption that Ft0 is an immersion. So p 6= qand W is closed and, from its own definition, W ∩∆ = ∅.

Hence, we can take a regular domain Ω ⊂M ×M such that

W ⊂ Ω ⊂ Ω ⊂M ×M \∆.

Let us define u : Ω× (0, t0) −→ R

u(p, q, t) := |F (p, t)− F (q, t)|2 .

As u|∂Ω × (0, t0) > σ > 0, for some12 σ, then we can guarantee theexistence of t1 ∈ (0, t0) such that u : Ω × (t1, t0) −→ R satisfies thehypotheses of Hamilton’s trick (Lemma 5.6). So, we have that

umin : (t1, t0) −→ R

is locally Lipschitz and

d

dtumin(t) =

∂

∂tu(p0, q0, t), t ∈ (t1, t0),

12recall that ∂Ω is compact


where (p0, q0) is any point where u(·, t) reaches its minimum. Butreasoning as in the proof of Theorem 5.8, we deduce that

∂

∂tu(p0, q0, t) ≥ 0, t ∈ (t1, t0).

This is absurd because umin(t0) = 0 and umin(t) > 0, for any t ∈ (t1, t0).This contradiction proves that t0 = T and concludes the proof.

6. Area Estimates and Monotonicity Formulas

Mean curvature flow can be also formulated in an integral way. Usingthis formulation, we will deduce area estimates and we focus our atten-tion on monotonicity formulas and its consequences. These formulasare considered the most important tools in the study of the formationand structure of singularities.

Throughout all this section we denote by (Mt)t∈I , where I is an openinterval of R (usually I = (0, T )), a family of smooth hypersurfacesmoving, properly embedded by the mean curvature. Recall that thisjust means that there is a family of smooth, proper, embeddings13

F : Mn × I → Rn+1, where:

(6.1)∂

∂tF (p, t) = ~H(p, t) ∀(p, t) ∈Mn × I,

As we already did in the previous section, we denote Mt := F (Mn, t).

To derive an integral version of the mean curvature flow is essentialto recall how they change the area elements of the hypersurfaces Mt.These are given by

dµt =√

det gijdµM

where dµM denotes the area form of M . This volume form determinesa measure on Mt which essentially coincides with the n- dimensionalHausdorff measure of Rn+1, H n, restricted to the hypersurface Mt.For details more details about integration on Riemannian manifold werecommend [Cha93,Mor00]. So, given a measurable function f : Mt →R, we denote:∫

Mt

f ≡∫Mt

f(x)dH n(x) =

∫M

f(F (p, t))dµt.

13Theorem 5.13 precisely says that if the initial hypersurface is compact andembedded, then they remain embedded along the flow.


The evolution of the area element under the mean curvature flow wasalready computed in (3.2), when we were considering the evolution ofthe intrinsic geometry

(6.2)∂

∂tdµt = −| ~H|2dµt

for all t ∈ I.If we consider a domain Ω ⊂ Rn+1 and f ∈ C1

0(Ω) (i.e., f is a C1

function with compact support), then

∂

∂t(f(F (p, t))dµt) = 〈Df,

∂

∂tF (p, t)〉dµt + f(F (p, t))

∂

∂tdµt =

using that F is a solution of (6.1) and (6.2), we deduce

= 〈Df, ~H〉dµt − f(F (p, t))| ~H|2dµt.

In this way we have proven the following:

Theorem 6.1 (Integral form for the Mean Curvature Flow,[Bra78]). Given (Mt)t∈I a mean curvature flow contained in Ω ⊂ Rn+1

we have

(6.3)d

dt

∫Mt

f =

∫Mt

(〈Df, ~H〉 − | ~H|2f

)for any t ∈ I and f ∈ C1

0(Ω).

If M is compact, we can obtain the following corollary

Corollary 6.2 (Area decreasing property). The mean curvatureflow of a compact hypersurface decreases the area. To be more precise,given a family (Mt)t∈I of compact solutions of the mean cuvature flowwe have that:

d

dtArea(Mt) = −

∫Mt

| ~H|2

for all t ∈ I.

Remark 6.3 (Brakke’s solutions). Theorem 6.1 can be used as adefinition of mean curvature flow, because any family of smooth, prop-erly embedded, hypersurfaces verifying (6.3) also is a solution in thesense of (6.1) (which is the one we have been considering throughoutthese notes). In fact, (6.3) is the motivation of the concept of “Brakke’s


solution”. Brakke defined the mean curvature flow for generalized hy-persurfaces (called integral varifolds in the language of Geometric Mea-sure Theory.)

Following the ideas of K. Ecker [Eck04], we will derive all the resultsof this section by substituting appropriate test functions in identity(6.3).

Definition 6.4 (Time-dependent test function). Let Ω be an openset in Rn+1, I an interval in R and φ ∈ C1(Ω × I). We say that φis a time-dependent test function if it satisfies φ(·, t) ∈ C2

0(Ω) and∂∂tφ(·, t) ∈ C0

0(Ω) for all t ∈ I.

Notice that if φ : Ω→ R is like in the previous definition, then φ and∂φ∂t

are integrable on Mt. Moreover, using the divergence theorem14 wehave that , for any t ∈ I,

(6.4)

∫Mt

∆Mtφ =

∫Mt

divMt Dφ+ 〈 ~H,Dφ〉 = 0.

For the sake of simplicity, in what follows we will write∫Mtφ instead

of∫Mtφ(·, t).

For a test function we have that:

(6.5)d

dt[φ(F (p, t), t) · dµt] =[〈Dφ, ~H〉+

∂φ

∂t(F (p, t), t)

]︸︷︷︸

= dφdt

· dµt − φ(F (p, t), t) · | ~H|2 · dµt

Hence, combining (6.1), (6.4) and (6.5) we immediately obtain the fol-lowing result.

Proposition 6.5 ([Bra78, Eck04]). Let (Mt)t∈I be a mean curvatureflow in Ω ⊂ Rn+1 and let φ be a time-dependent test function on Ω× I.

14Here we use that:

∆Mtf = divMt

Df + 〈 ~H,Df〉,

where f : Ω× I → R is sufficiently regular. D denotes the gradient in Rn+1.


Then, the following equalities hold:

(6.6)d

dt

∫Mt

φ =

∫Mt

dφ

dt− |H|2φ =

∫Mt

∂φ

∂t+ 〈 ~H,Dφ〉 − |H|2φ,

(6.7)d

dt

∫Mt

φ =∫Mt

∂φ

∂t− divMt Dφ− |H|2φ =

∫Mt

(d

dt−∆Mt

)φ− |H|2φ

and

(6.8)d

dt

∫Mt

φ =

∫Mt

(d

dt+ ∆Mt

)φ− |H|2φ.

As an immediate consequence of this proposition we conclude that ina certain sense (specified with an additional hypothesis in the followingresult) the area of the solutions of mean curvature flow also decreaseslocally.

Corollary 6.6 (Local area decay, [Bra78, Eck04]). If the test func-tion φ verifies

(6.9)

(d

dt−∆Mt

)φ =

∂φ

∂t− divMt Dφ ≤ 0,

then we have:

(6.10)d

dt

∫Mt

φ ≤ −∫Mt

| ~H|2φ

for each t ∈ I.

Definition 6.7. Spherically shrinking test functions. An importantclass of test functions is the following: given ρ > 0 and (x, t) ∈ Rn+1×Iwe consider15

ϕρ(x, t) =

(1− |x|

2 + 2nt

ρ2

)3

+

and its translations:

ϕ(x0,t0),ρ(x, t) = ϕρ(x− x0, t− t0).

Lemma 6.8. The function ϕ(x0,t0),ρ satisfies:

15As usual f+ means f+(x) := max(f(x), 0).


Figure 4. The evolution of ϕρ(·, t), as t→ 0.

(i) spt(ϕ(x0,t0),ρ(·, t)) ⊂ B(x0,√ρ2 − 2n(t0 − t)).

(ii) The maximum of ϕ(x0,t0),ρ(·, t) is reached at x = x0, where

ϕ(x0,t0),ρ(x0, t) =ρ2 − 2n(t− t0)

ρ2.

(iii) Moreover, the following inequality holds:

(6.11)

(d

dt−∆Mt

)ϕ(x0,t0),ρ ≤ 0.

Proof. The first two properties are trivial. Regarding item (iii), we aregoing to write ϕ(x0,t0),ρ = g u, where g : R −→ R is given by

g(r) =

(1− r

ρ2

)3

+

,

and u : Rn+1 × (−∞, t0) −→ R

u(x, t) = |x− x0|2 + 2n(t− t0).


If we compute ∆Mtu = divMt Du + 〈 ~H,Du〉. It is clear that Du =2(x− x0), so

∆Mtu = 2(n+ 〈 ~H, x− x0〉

).

On the other hand, taking into account that Mt is solution of the MCFequation, we have

d

dtu = 2〈 ~H, x− x0〉+ 2n.

From the last two equalities, we get

(6.12)

(d

dt−∆Mt

)u = 0.

Taking into account that

∆Mt(g(u)) = g′(u) ·∆Mtu+ g′′(u) · |∇u|2,and

d

dtg(u) = g′(u) · d

dtu.

Therefore, using (6.12), we get(d

dt−∆Mt

)g(u) =

g′(u)

(d

dt−∆Mt

)u− g′′(u) · |∇u|2 = −g′′(u) · |∇u|2.

Finally, notice that

g′′(r) =6

ρ4

(1− r

ρ2

)+

≥ 0,

which concludes the proof.

Using these test functions, the following local estimate of the area isobtained:

Proposition 6.9 (Local area estimate, [Bra78, Eck95]). Let Mt bea smooth, properly embedded, solution of the mean curvature flow in

B(x0, ρ)× (t0 − ρ2

8n, t0). Then we have

Area(Mt ∩B(x0,

ρ

2))

+

∫ t

t0− ρ2

8n

∫Mt∩B(x0,

ρ2

)

| ~H|2

≤ 8 Area(M

t0− ρ2

8n

∩B(x0, ρ))


for all t ∈ [t0 − ρ2

8n, t0].

Proof. Take φ = ϕ(x0,t0−ρ2/(8n)),ρ. So, we integrate over the interval

(t0 − ρ2

8n, t) the inequality given by Corollary 6.6 and obtain∫

Mt

φ−∫Mt0−

ρ2

8n

φ ≤ −∫ t

t0− ρ2

8n

∫Mt∩B(x0,

ρ2

)

| ~H|2φ,

or equivalently,∫Mt

φ+

∫ t

t0− ρ2

8n

∫Mt∩B(x0,

ρ2

)

| ~H|2φ,≤∫Mt0−

ρ2

8n

φ

Now, note that φ(x, t) ≥ 1/8, whenever |x − x0| ≤ ρ/2 and t ≤t0 + ρ2/(8n). So, we use this lower bound on the left-hand side ofthe inequality. On the right-hand side, we apply that φ ≤ 1. Thiscompletes the proof.

We would like to point out that, using Corollary 6.6, we can deducea “comparison principle” with the sphere.

Proposition 6.10. If we have a mean curvature flow Mt, t ∈ [0, T ),satisfying

Area(M0 ∩B(0, ρ)) = 0,

then

Area(Mt ∩B(0,√ρ2 − 2nt)) = 0, for all t <

ρ2

2n.

Proof. By definition the function ϕρ is non-negative. It also follows im-mediately from its definition that its support is the ball with center atthe origin and radius

√ρ2 − 2nt, i.e. B(0,

√ρ2 − 2nt). Then, the inte-

gral of ϕρ on Mt is the same as integrating it over Mt∩B(0,√ρ2 − 2nt)

. Also as ϕρ is non-negative, so its integral is also non-negative. Then,in this case, Corollary 6.6 says to us that

∫Mtϕρ is not decreasing in t

(and it is non-negative). Therefore, we conclude that if at the initialtime this function is zero, it must remain zero.

Remark 6.11. If Mt is compact, then we use ϕ(x, t) = (−ϕρ(x, t))3+

to deduce

Area(M0 \B(0, ρ)) = 0⇒ Area(Mt \B(0,√ρ2 − 2nt)) = 0,

for t < ρ2

2n.


One can check that the area estimate provided by Proposition 6.9 cannot be used to bound the ratio

Area(Mt ∩B(x0, ρ))

ρn

independently of ρ in a time interval of length proportional to ρ2; toestablish such an estimate valid in a longer time interval an more accu-rate result is needed (see [Eck04, p. 75].) In order to do this we needthe following definition:

Definition 6.12 (Backward Heat Kernels). Given x ∈ Rn+1 yt < 0,

Φ(x, t) =1

(−4πt)n2

e|x|24t

and its translations

Φ(x0,t0)(x, t) = Φ(x− x0, t− t0) =1(

4π(t0 − t))n

2

e− |x−x0|

2

4(t0−t)

where x0 ∈ Rn+1 and t < t0.

Remark 6.13. Note that Φ(x, t) = 1√−4πt

Ψ(x, t), where Ψ is the stan-

dard heat kernel in Euclidean space. Then, we have that:

(6.13)

(∂

∂t+ ∆Rn+1

)Φ(x0,t0) =

1

2 (t− t0)Φ(x0,t0).

Bearing in mind the similarities of (2.2) with the standard heat equa-tion, it makes sense that the function Φ can be useful to study themean curvature flow.

Lemma 6.14. In the previous setting, the following equality holds:

∂Φ

∂t+ divMt DΦ +

|∇⊥Φ|2

Φ= 0.

Proof. Recall that divMt DΦ = ∆Rn+1Φ−D2Φ(ν, ν), where D2Φ is theEuclidean Hessian of Φ and ν is the Gauß map. On the other hand,from its definition we trivially deduce that, in general,

DΦ(x0,t0) =Φ(x0,t0)

2(t− t0)· (x− x0),

and so

∇⊥Φ(x0,t0) =Φ(x0,t0)

2(t− t0)· (x− x0)⊥,(6.14)

|∇⊥Φ(x0,t0)| =Φ(x0,t0)

2(t− t0)· 〈x− x0, ν〉.(6.15)


So, using 6.13 and (6.15) (for t0 = 0 and x0 = 0), we deduce that

(6.16)∂Φ

∂t+ divMt DΦ +

|∇⊥Φ|2

Φ=

∂Φ

∂t+ ∆Rn+1Φ−D2Φ(ν, ν) +

〈x, ν〉2

4t2Φ =

1

2tΦ−D2Φ(ν, ν) +

〈x, ν〉2

4t2Φ.

If we compute the Hessian term, we get:

D2Φ(ν, ν) =1

4t2〈x, ν〉2Φ +

1

2t〈ν, ν〉Φ.

Using that ν is unitary and substituting in (6.16), we conclude theproof.

The next result is a monotonicity formula proved by Huisken in[Hui90]. It deals with the monotonic behaviour of the integral overMt of the backward heat kernel Φ(x0,t0)(x, t).

Theorem 6.15 (Monotonicity formula). Consider (x0, t0) ∈ Rn+1×R and (Mt)t∈I a smooth solution of the mean curvature flow satisfying∫Mt

Φ(x0,t0) <∞ ∀t ∈ I with t < t0. Then, for t < t0, we have

d

dt

∫Mt

Φ(x0,t0) = −∫Mt

∣∣∣∣ ~H − ∇⊥Φ(x0,t0)

Φ(x0,t0)

∣∣∣∣2Φ(x0,t0).

In particular,∫Mt

Φ(x0,t0) is decrasing for any t < t0.Moreover, for x ∈Mt and t < t0 the following equality holds:

~H(x)−∇⊥Φ(x0,t0)(x, t)

Φ(x0,t0)(x, t)= ~H(x)− (x− x0)⊥

2(t− t0).

Proof. From now on, and for simplicity, we will write Φ0 instead ofΦ(x0,t0).

So, with this notation, we have:(d

dt+ ∆Mt

)Φ0 =

∂Φ0

∂t+ 〈 ~H,DΦ0〉+ divMt DΦ0 + 〈 ~H,DΦ0〉 =

(recall that, using Gauß equation, we have divMt DΦ0 = ∆MtΦ0 −〈 ~H,DΦ0〉)

=∂Φ0

∂t+ divMt DΦ0 + 2 〈 ~H,DΦ0〉.


Now we use that 〈 ~H,DΦ0〉 = 〈 ~H,∇⊥Φ0〉 and so,(d

dt+ ∆Mt

)Φ0 =

∂Φ0

∂t+ divMt DΦ0 + 2 〈 ~H,∇⊥Φ0〉−∣∣∣∣ ~H − ∇⊥Φ0

Φ0

∣∣∣∣2 Φ0 +

∣∣∣∣ ~H − ∇⊥Φ0

Φ0

∣∣∣∣2 Φ0 =

=∂Φ0

∂t+ divMt DΦ0 +

|∇⊥Φ0|2

Φ0

−∣∣∣∣ ~H − ∇⊥Φ0

Φ0

∣∣∣∣2 Φ0 + | ~H|2Φ0.

Using Lemma 6.14, we substitute to obtain that

(6.17)

(d

dt+ ∆Mt

)Φ0 = −

∣∣∣∣ ~H − ∇⊥Φ0

Φ0

∣∣∣∣2 Φ0 + | ~H|2Φ0.

If Mt is compact, we use (6.8) to finish.

In the general case, we are going to use a test function φ, whoseproperties will be determined later.

(6.18)d

dt

(∫Mt

φ · Φ0

)=

∫Mt

(dφ

dtΦ0 +

dΦ0

dtφ− | ~H|2 · φ · Φ0

)Now, we use that

divMt(φ∇Φ0 − Φ0∇φ) = φ∆MtΦ0 − Φ0∆Mtφ.

Integrating over Mt, and using that φ has compact support, we obtain(applying the divergence theorem)

(6.19)

∫Mt

φ∆MtΦ0 − Φ0∆Mtφ = 0.

Adding (6.18) + (6.19) we get

d

dt

(∫Mt

φ · Φ0

)=∫

Mt

Φ0

(d

dt−∆Mt

)φ+ φ

[(d

dt+ ∆Mt

)Φ0 − | ~H|2Φ0

]Substituting (6.17) in the previous equality, it becomes:

(6.20)d

dt

(∫Mt

φ · Φ0

)=∫Mt

Φ0

(d

dt−∆Mt

)φ−

∣∣∣∣ ~H − ∇⊥Φ0

Φ0

∣∣∣∣2 Φ0 φ.

At this point, we are going to make our choice of φ. Fix R > 0, thenwe take φ a smooth function satisfying:


• χB(x0,R) ≤ φ ≤ χB(x0,2R).• R|Dφ|+R2|D2φ| ≤ C0, where C0 is a positive constant.

For the existence of such a function see [Ilm95]. Hence, for this partic-ular φ, we have: ∣∣∣∣( d

dt−∆Mt

)φ

∣∣∣∣ ≤ κ

R2,

for κ a constant depending on n and C0. At this point, we can applythe theorems of convergence under the integral sign and take limit, asR→ +∞. The first integral of the right-hand side goes to zero. Takinginto account that limR→∞ φ ≡ 1, we conclude the proof.

Roughly speaking, the monotonicity formula means that the area ofa hypersurface which moves by the mean curvature near any point isnon-increasing on any scale. Actually, it strictly decreases unless thehypersurface is homothetically shrinking around the point x0.

Remark 6.16. Notice that the assumption∫Mt

Φ(x0,t0) <∞

implies that Mt has locally finite H n-measure, which is all we need forthe integral formulation of the mean curvature flow (6.3). A solution(Mt)t∈I of the mean curvature flow (6.1) satisfies this assumption forany t ∈ I if we require that the flow has polynomial area-growth whenyou intersect the flow with sufficiently large balls, that is to say

Area(Mt ∩B(x0, R)) ≤ ARp, for all t ∈ I and R >> 0,

(see [Eck04, Lemma C.3].)

This inequality can be obtained due to an appropriate condition oninitial hypersurface. Indeed, if for instance I = (0, T ) and M0 satisfies

Area(M0 ∩B(x0, R)) ≤ ARp

for all R ≥ R0, for certain R0 big enough, then the local estimation ofProposition 6.9 implies

Area(Mt ∩B(x0, R)) ≤ 2p+3ARp

for each t ∈ (0, T ) as long as R ≥ max√

2nT ,R0/2.

The proof given in [Eck04] of the monotonicity formula also producesthe following extension involving weight functions that (jointly with


their derivatives) are integrable on Mt with respect to Φ(x0,t0). In par-ticular, these functions with compact support defined on a properlyembedded solution (6.1) form an important family. This extension isthe base of a local version of the formula of monotonicity ( Proposition6.23), which is a fundamental tool in regularity theory in the first sin-gular moment. Below we state the specific result.

Theorem 6.17 (Weighted Monotonicity Formula). Let (Mt)t∈Ibe a family of surfaces moving by the mean curvature and fix (x0, t0) ∈Rn+1×R. Assume f is a smooth function (probably depending on time)define on (Mt)t∈I satisfying:∫

Mt

(|f |+ |∂f

∂t|+ |Df |+ |D2f |

)Φ(x0,t0) <∞

for all t ∈ I con t < t0. Then, for t < t0, we have

d

dt

∫Mt

fΦ(x0,t0) =

∫Mt

((d

dt−∆Mt

)f −

∣∣∣∣ ~H − ∇⊥Φ(x0,t0)

Φ(x0,t0)

∣∣∣∣2f)Φ(x0,t0).

Moreover, if f is non-negative and verifies:(d

dt−∆Mt

)f ≤ 0,

thend

dt

∫Mt

fΦ(x0,t0) ≤ 0.

Proof. The demonstration of this theorem follows the same steps as inTheorem 6.15. The only difference is that we use the function f · φinstead of φ as a test function in formula (6.20).

Remark 6.18. Representation formula for the standard heat equation.Let Mt = Rn for any t < t0 and consider f = f(x, t) a solution of thestandard heat equation in Rn. Theorem 6.17 yields

d

dt

∫Rnf(x, t)Φ(x0,t0)(x, t)dx = 0

for all t < t0. Indeed,

d

dt

∫RnfΦ(x0,t0) =

∫Rn

((d

dt−∆Mt

)f −

∣∣∣∣ ~H − ∇⊥Φ(x0,t0)

Φ(x0,t0)

∣∣∣∣2f)Φ(x0,t0) =

=

∫Rn

(0−

∣∣∣∣0− ∇⊥Φ(x0,t0)

Φ(x0,t0)

∣∣∣∣2f)Φ(x0,t0) = 0


where we are using that

(ddt− ∆Mt

)f = 0, that ~H = 0 and that

∇⊥Φ(x0,t0) = 0.Therefore

∫Rn f(x, t)Φ(x0,t0)(x, t)dx does not depend on t for t < t0. On

the other hand, as f is continuous, we have that

limtt0

∫Rnf(x, t)Φ(x0,t0)(x, t)dx = f(x0, t0),

Now we will discuss some consequences of the monotonicity formula.

Theorem 6.19 (Upper Bound for the Area Ratio). Consider a smooth,properly embedded MCF, Mt, in B(x0, r0)× (t0 − r2

0, t0). Then

supt∈(t0−r20 ,t0)

Area(Mt ∩B(x0, r))

rn≤

κ(n) ·Area(Mt0−r20/(2(2n+1)) ∩B(x0, r0))

rn0

for any r ∈

(0,

r0√2(2n+ 1)

).

Proof. Take a =√

1/(2(2n+ 1)) and consider

f = ϕ(x0,t0−a2r20),r0 .

Fix r ∈ (0, a r0). Since we have that(d

dt−∆Mt

)f ≤ 0,

then we can apply the Weighted Monotonicity Formula to thefunction f and the point (x0, t0 + r2) ∈ Rn+1 × R.

So, for t ∈ (t0 − r20, t0), we have:∫

Mt

Φ(x0,t0+r2) · ϕ(x0,t0−a2r20),r0 dµt ≤∫Mt0−a2r20

Φ(x0,t0+r2) · ϕ(x0,t0−a2r20),r0 dµt

For the right-hand side we use that:

• spt(ϕ(x0,t0−a2r20),r0) ⊆ B(x0, r0).

• |ϕ(x0,t0−a2r20),r0| ≤ 1 and |Φ(x0,t0+r2)| ≤1

anrn0


For the left-hand side we use that, for r ∈ (0, a r0), one has:

ϕ(x0,t0−a2r20),r0 ≥1

8and

Φ(x0,t0+r2) ≥1

e1/4(8π)n/2rn

on B(x0, r)× (t0 − r2, t0). This completes the proof.

In order to refer more easily to the solutions of the mean curvatureflow we will often denote M = (Mt)t∈I .

An immediate consequence of the non-positivity of the right-hand sideof the Monotonicity Formula (Theorem 6.15) is the following proposi-tion (see [Eck04]):

Proposition 6.20 (Gaussian Density). Let M = (Mt)t∈(t1,t0) be asolution of the MCF satisfying the assumptions of Theorem 6.15. Thenfor any x0 ∈ Rn+1 the Gaussian density

(6.21) Θ(M , x0, t0) = limtt0

∫Mt

Φ(x0,t0)

exists.The solutions of the mean curvature flow which are homothetically con-tracting with respect to (x0, t0), i.e., satisfying (see [Eck04, p. 10-13])

~H(x) =(x− x0)⊥

2(t− t0)

for all x ∈Mt y t < t0, can be also characterized by the property:

(6.22) Θ(M , x0, t0) =

∫Mt

Φ(x0,t0)

for all t < t0.

Remark 6.21. It is clear from its definition that, for any t ∈ (t1, t0),

Θ(M , x0, t0) ≤∫Mt

Φ(x0,t0).

Example 6.22. (A) Our first example is the case of hyperplane P ⊂Rn+1. Taking into account that:• the Hausdorff measure is invariant under isometries of Eu-

clidean space,


• the measure H m in Rm coincides with the standard Lebesguemeasure m-dimensional,• the Riemann integral of the heat kernel on Rn is 1,

then

(6.23)

∫P

Φ(x0,t0)(x, t)dHn(x) =

∫Rn

Φ(x)dx = 1

for all t < t0. Therefore,

(6.24) Θ(M , x0, t0) = limtt0

∫Mt

Φ(x0,t0) = limtt0

1 = 1

where M = P because a hyperplane is minimal and therefore it isstationary.

(B) A. Stone computed in [Sto94] the density for other flows like shrink-ing spheres and cylinders. We consider Mt := Sn−mr(t) × Rm, r(t) =√

2(m− n)t, t < 0. It is clear that M−1/2 = Sn−m√n−m × Rm, so by

(6.22) in the above proposition we have

Θ(M , 0, 0) =

∫M−1/2

e−|x|2/2

(2π)n/2dµ =

(n− k2π e

)n−k2

· Area(Sn−m√

n−m

)> 1

Thus, for the surfaces in Euclidean 3-space we have that Θ(M , 0, 0) =

4/ e in the case of cylinders and Θ(M , 0, 0) =√

(2π)/ e then Mconsists of concentric spheres.

6.1. Parabolic rescaling. In the general case, the density value canbe calculated using parabolic scale changes of the solution M = (Mt)t∈Iin the way we are going to describe in this subsection. We considerx ∈Mt and make the following change of variable:

x = λy + x0

t = λ2s+ t0

where λ > 0 and s < 0. Then

y ∈ 1

λ(Mλ2s+t0 − x0) ≡M (x0,t0),λ

s

and for a fix λ > 0

(M (x0,t0),λs )s<0

is solution of the MCF equation.This will show at the end of this sectionfor not diverting our attention from the main objective of studying theGaussian density following a parabolic scaling.


Figure 5. Shrinking cylinders.

The integral of the Gaussian density can be calculated by applying achange of variable:∫

Mt

Φ(x0,t0) =

∫Mt

1

[4π(t0 − t)]n/2e− |x−x0|

2

4(t0−t) dH n(x) =

=

∫1λ

(Mλ2s+t0−x0)

1

(−4πs)n/2e|y|24s dH n(y) =

∫M

(x0,t0),λs

Φ.

Therefore for any s < 0,

(6.25) Θ(M , x0, t0) = limtt0

∫Mt

Φ(x0,t0) = limλ0

∫M

(x0,t0),λs

Φ.


In case the point (x0, t0) ∈ Rn+1 × R verifies x0 ∈Mt0 , then

limλ0

M (x0,t0),λs = Tx0Mt0

for all s < 0 (smoothly on compact sets). Taking this into account andusing (6.25) and (6.24), we get

(6.26) Θ(M , x0, t0) =

∫Tx0Mt0

Φ(y, s)dH n(y) = 1.

Finally, as we promised, we want to see that (M(x0,t0),λs )s<0 is a solu-

tion of the MCF equation. Indeed, we know that Mt moves by the meancurvature, that is, the re exits a smooth embedding Ft = F (·, t) : M →Rn+1 with Mt = Ft(M) and such that ∂F

∂t(x, t) = ~H(x, t) ∀t ∈ I.

To check that (M(x0,t0),λs )s∈J is also solution it suffices to prove that

Gs = G(·, s) : M → Rn+1 given by

G(p, s) =1

λF (p, λ2s+ t0)

verifies:∂

∂sG(p, s) = ~HG(p, s),

where ~HG is the mean curvature of G, which is ~HG = λ · ~H. Hence,the equality we are looking for is just an easy consequence of the chainrule.

6.2. Local monotonicity formula. In this subsection we are goingto use again the test functions:

ϕ(x0,t0),ρ(x, t) =

(1− |x− x0|2 + 2n(t− t0)

ρ2

)3

+

that were already introduced in Remark 6.7. We used them in theproof of Proposition 6.9 (local area bound.)

Take (Mt)t∈(t1,t0) a family of hypersurfaces moving by the mean cur-vature inside a domain U ⊂ Rn+1. Fix x0 ∈ U , there exists ρ0 > 0 sothat

B(x0,√

1 + 2n ρ0)× (t0 − ρ20, t0) ⊂ U × (t1, t0).

For each ρ ∈ (0, ρ0) and t ∈ (t0 − ρ20, t0) we have that that the support

of our test function satisfies:

(6.27) sptϕ(x0,t0),ρ(·, t) ⊂ B(x0,√

1 + 2n ρ0) ⊂ U.


Proposition 6.23 (Local Monotonicity Formula). Let (Mt)t∈(t1,t0)

a smooth mean curvature flow in U ⊂ Rn+1. Then for each x0 ∈ Uthere exists a ρ0 ∈ (0,

√t0 − t1) such that for all ρ ∈ (0, ρ0] and t ∈

(t0 − ρ2, t0)

sptϕ(x0,t0),ρ(·, t) ⊂ U

and(6.28)

d

dt

∫Mt

Φ(x0,t0)ϕ(x0,t0),ρ ≤ −∫Mt

∣∣∣∣ ~H(x)− (x− x0)⊥

2(t− t0)

∣∣∣∣2Φ(x0,t0)ϕ(x0,t0),ρ .

Since the right-hand side of (6.28) is negative and ϕ(x0,t0),ρ(x0, t0) = 1for each ρ ∈ (0, ρ0], then the following local Gaussian density

(6.29) Θ(M , x0, t0) = limtt0

∫Mt

Φ(x0,t0)ϕ(x0,t0),ρ

is well defined, is independent of ρ and for global solutions of MCFequation it coincides with the density defined in Proposition 6.20.Furthermore, for all t ∈ (t0 − ρ2, t0)

(6.30) Θ(M , x0, t0) ≤∫Mt

Φ(x0,t0)ϕ(x0,t0),ρ .

Proof. In order to get 6.28 we only have to apply (6.20) for φ = ϕ(x0,t0),ρ,taking into account that ϕ(x0,t0),ρ satisfies (6.11).

For the second part of the proof we are going to use some ideas dueto B. White [Whi97]. For x0 ∈ U and ρ > 0 such that

B(x0, 2ρ)× (t0 −ρ2

2n, t0) ⊂ U × (t1, t0).

By Proposition 6.9 we deduce

sup(t0− ρ

2

2n,t0)

Area (Mt ∩B(x0, ρ)) ≤ 8 Area(M

t0− ρ2

2n

∩B(x0, 2 ρ))≡ k0.

Consider ψρ ∈ C20(B(x0, ρ)) satisfying: χB(x0,ρ/2) ≤ ψρ ≤ χB(x0,ρ) and

|D2ψρ| ≤ k1, where c1 depends on ρ. Then∣∣∣∣( d

dt−∆Mt

)ψρ

∣∣∣∣ ≤ k2 · χB(x0,ρ)−B(x0,ρ/2),

for a suitable constant k2 depending on n and k1. On B(x0, ρ) −B(x0, ρ/2) we have Φ(x0,t0) ≤ k3, where k3 depends on n and ρ. Then


the weighted monotonicity formula gives that

d

dt

∫Mt

ψρΦ(x0,t0) ≤ k4,

where k4 depends on the previous constants. So, the function t 7→∫MtψρΦ(x0,t0) − c4 t is non-increasing and therefore the following limit

exists

limtt0

∫Mt

ψρΦ(x0,t0).

It is possible to check that this limit is independent of ρ [Whi97]. andhence it does not depend on the particular function ψρ which is con-stantly 1 in an neighborhood of x0. So, (6.29) makes sense. Inequality(6.30) is consequence of the monotonic behavior of the integral.

Remark 6.24 (Local monotonicity under re-scaling). Changingthe scale in U × (t1, t0) as we made in Remark 6.21, then a smooth,properly embedded, mean curvature flow (Mt)t∈(t1,t0) becomes

(M (x0,t0),λs )

s∈(λ−2(t1−t0),0

) in λ−1(U − x0)× (λ−2(t1 − t0), 0).

If ρ0 is such that B(x0,√

1 + 2nρ0) ⊂ U , then for any ρ ∈ (0, ρ0] wehave:

(6.31) Θ(M , x0, t0) = limλ0

∫M

(x0,t0),λs

Φ(y, s)ϕλ−1ρ(y, s)dHn(y)

for all s < 0, where

ϕλ−1(y, s) =

(1− λ2 |y|2 + 2ns

ρ2

)3

+

is the re-scaled test function.In particular, as we made in Remark 6.21, we can deduce

(6.32) Θ(M , x0, t0) = 1

for all x0 ∈Mt0 .

One of the main applications of the local monotonicity is the studyof the limits of mean curvature flow after changes in scale. The nextresult, which concludes this section, goes in this direction. It is an im-mediate consequence of Propositions 6.20 and 6.23 and Remark 6.24.

Proposition 6.25 (Tangent flows/ Parabolic Blow-ups). Let M =(Mt)t∈(t1,t0) be a smooth, properly embedded, mean curvature flow in adomain U ⊂ Rn+1. Let x0 ∈ U and assume that for a given sequence


λjj∈N 0 the hypersurfaces (M(x0,t0),λjs ) in λ−1

j (U−x0)×(λ−2j (t1−

t0), 0) obtained as in Remark 6.21 converge16, smoothly on compactsets of Rn+1, to a properly embedded solution of the MCF equationM ′ = (M ′

s)s<0. Then M ′ satisfies

(6.33) Θ(M ′, 0, 0) =

∫M ′s

Φ = Θ(M , x0, t0)

for all s < 0. Therefore, from Theorem 6.15, we deduce

(6.34) ~H(y) =y⊥

2s

for all y ∈M ′s and s < 0. Moreover, this means that

(6.35) M ′s =√−s ·M ′

−1

for each s < 0.

Definition 6.26. The limit M ′ = (M ′s)s<0 is called tangent flow or

parabolic “blow-up” of M in (x0, t0).

It is posible to show (6.35) from (6.34), for instance by proving di-rectly from the definition of MCF that the hypersurfaces (M ′

s) verify

∂

∂s

(F (φ(p, s), s)√

−s

)= 0

where φ(·, s) : M →M is a family of diffeomorphisms satisfying

dF

(∂φ

∂s

)= −

(∂F

∂s

)>.

16Given (Mj)j∈N ⊂ U and M ⊂ U properly embedded hypersurfaces inU ⊂ Rn+1, we say that (Mj , gj)j∈N smoothly converge to (M, g) in U if M isthe pointwise limit of (Mj)j∈Nand for all p ∈M there exist r, ε > 0 such that

(1) M ∩W (p, r, ε) can be writen as the graph of a function u : D(p, r)→ R.(2) For each j (large enough) the hypersurface Mj ∩ W (p, r, ε) can be also

written as the graph of a function uj : D(p, r)→ R and uj converges tou in the topology of Ck convergence on compact subsets of D(p, r), for anyk ∈ N,

Here we have used the following notation: D(p, r) = p + v : v ∈ TpM, |v| < rmeans the tangent disk of radius r > 0 and W (p, r) = q+tν(q) : q ∈ D(p, r), t ∈ Ris the solid cylinder of radius r around the normal line determined by the Gaußmap of M at p; ν(p). Inside W (p, r), given ε > 0, we take the compact cylinderW (p, r, ε) = q + tN(q) : q ∈ D(p, r), |t| < ε.


This means that F (φ(p,s),s)√−s is a family of embeddings that does not

depend on s. As at the instant s = −1 we know that

F (φ(·,−1),−1)√−(−1)

= F (φ(·,−1),−1) = M ′−1,

then F (φ(·,s),s)√−s = M ′

−1 for all s < 0, that is, M ′s =

√−sM ′

−1, as we

wanted to prove.

7. Some Remarks About Singularities

Throughout this section, we consider a compact initial hypersurfaceM . Consider T maximal such that a smooth solution of the MCFF : M × [0, T )→ Rn+1 as in Theorem 2.6 exists. Then the embeddingvector F is uniformly bounded according to Corollary 5.10. Then somespatial derivatives of the embedding Ft have to become unbounded ast T . Otherwise, we could apply Arzela-Ascoli Theorem and obtaina smooth limit hypersurface, MT , such that Mt converges smoothly toMT as t T . This is impossible because, in such a case, we couldapply Theorem 2.6 to re-start the flow. In this way, we could extendthe flow smoothly all the way up to T +ε, for some ε > 0 small enough,contradicting the maximality of T . In particular, we have that |h|2 isnot bounded, when we approach the maximal time T .

We would like to say more about the “blowing-up” of the norm of h,as t T. Recall that , according to (3.9), the evolution equation for|h|2 is

∂

∂t|h|2 = ∆|h|2 − 2

∣∣∇h∣∣2 + 2|h|4.

Label|h|2max := max

Mt

|h|2(·, t).

Using Hamilton’s trick (Lemma 5.6) we deduce that |h|2max is locallyLipschitz and that

d

dt|h|2max(t0) =

∂

∂t|h|2(p0, t0),

where p0 is any point where |h|2(·, t0) reaches its maximum. Thus,using the above expression, we have

d

dt|h|2max(t0) =

∂

∂t|h|2(p0, t0) =

∆|h|2(p0, t0)− 2∣∣∇h(p0, t0)

∣∣2 + 2|h|4(p0, t0)


It is well known that the Hessian of |h| is negative semi-definite atany maximum. In particular the Laplacian of |h| at these points isnon-positive. Hence,

d

dt|h|2max(t0) ≤ 2|h|4(p0, t0) ≤ 2|h|4max(t0).

Notice that |h|2max is always positive, otherwise at some instant t wewould have that h ≡ 0, along Mt, which would imply that Mt is ahyperplane Rn+1, which is contrary to the fact that the initial data isa compact hypersurface.So, one can prove that 1/|h|2max is locally Lipschitz. Then the previousinequality allows us to deduce that:

− d

dt

(1

|h|2max

)≤ 2, a.e. in t ∈ [0, T ).

Integrating (respect to time) in any sub-interval [t, s] ⊂ [0, T ) we get

1

|h(·, t)|2max

− 1

|h(·, s)|2max

≤ 2(s− t).

As h is not bounded as to tends to T , then there exists a time sequencesi T such that

|h(·, si)|2max → +∞.Substituting s = si in the above inequality and taking limit, as i→∞,we get

1

|h(·, t)|2max

≤ 2(T − t).

We collect all this information in the next proposition.

Proposition 7.1. Consider the mean curvature flow for compact ini-tial hypersurface M . If T is the maximal time of existence, then thefollowing lower bound holds

maxp∈M|h(p, t)| ≥ 1√

2(T − t)for all t ∈ [0, T ).In particular,

limt→T

maxp∈M|h(p, t)| = +∞.

Definition 7.2. When this happens we say that T is singular time forthe mean curvature flow.


So we have the following improved version of Theorem 2.6:

Theorem 7.3. Given a compact, immersed hypersurface M in Rn+1

then there exists a unique mean curvature flow defined on a maximalinterval [0, Tmax).Moreover, Tmax is finite and

maxp∈M|h(p, t)| ≥ 1√

2(Tmax − t)for each t ∈ [0, Tmax).

Remark 7.4. From the above proposition, we deduce the followingestimate for the maximal time of existence of flow:

Tmax ≥1

2|h(·, 0)|2max

.

Definition 7.5. Let T be the maximal time of existence of the meancurvature flow. If there is a constant C > 1 such that

maxp∈M|h(p, t)| ≤ C√

2(T − t),

then we say that the flow develops a Type I singularity at instant T .Otherwise, that is, if

lim supt→T

maxp∈M|h(p, t)|

√(T − t) = +∞,

we say that is a Type II singularity.

We conclude this brief section by pointing out that there have beensubstantial breakthroughs in the study and understanding of the sin-gularities of type I, whereas type II singularities have been much moredifficult to study. This seems reasonable since, according to the abovedefinition and the results we have seen, the singularities of type I arethose for which has the best possible control of “blow-up” of the secondfundamental form.

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Francisco MartınDepartmento de Geometrıa y TopologıaUniversidad de Granada18071 Granada, SpainE-mail address: [email protected]

Jesus PerezDepartmento de Geometrıa y TopologıaUniversidad de Granada18071 Granada, SpainE-mail address: [email protected]

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