the zoo of curve shortening solitons in rnangenent/preprints/ima-zootalk3.pdf · the zoo of curve...

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......The Zoo of Curve Shortening Solitons in Rn

Sigurd Angenent, Dylan&Steve Altschuler, Lani WuUW Madison, St.Mark’s School of Texas, UT Southwestern&UCSF

February 12, 2014

IMA talk the soliton zoo

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.. Curve Shortening in Rn

A family of space curves X = X(t, ξ), evolves by CurveShortening if

Xt = Xss + λXs,

for some function λ = λ(t, ξ).

Equivalently, X evolves by CS if

X⊥t = Xss,

where Xt = X⊥t + λXs is the decompo-sition of Xt into normal and tangentialcomponents.

Xt⊥

Xt

X(t)

X(t+dt)


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.. Solitons

Curve Shortening is invariant under a large number oftransformations of space-time, namely

Translations (t, X) 7→ (t + τ, X + a), τ ∈ R, a ∈ Rn.Rotations (t, X) 7→ (t, RX), R ∈ SO(n, R)

Parabolic Dilation (t, X) 7→ (σ2t, σX), σ ∈ R+

The full symmetry group G consists of combinations of thesetransformations,

g(t, x) =(σ2t + τ, σRx + a

)


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.. Solitons


Translations (t, X) 7→ (t + τ, X + a), τ ∈ R, a ∈ Rn.

Rotations (t, X) 7→ (t, RX), R ∈ SO(n, R)Parabolic Dilation (t, X) 7→ (σ2t, σX), σ ∈ R+


g(t, x) =(σ2t + τ, σRx + a

)


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.. Solitons


Translations (t, X) 7→ (t + τ, X + a), τ ∈ R, a ∈ Rn.Rotations (t, X) 7→ (t, RX), R ∈ SO(n, R)

Parabolic Dilation (t, X) 7→ (σ2t, σX), σ ∈ R+


g(t, x) =(σ2t + τ, σRx + a

)


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Self similar evolutionsDefinition: Solitons are solutions of Curve Shortening that are invariant under a oneparameter subgroup of symmetries {gθ : θ ∈ R} ⊂ G.

Most Curve Shortening solitons are of the form

X(t) = σ(t)R(t)C + a(t)

for some fixed space curve C that satisfies

Css = (α + A)C + v + λ(s)Cs

for some real valued function λ, where

α ∈ R is the expansion/shrinking rate,A = −AT ∈ so(n, R) generates the rotations R(t),v is the translation velocity


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X(t) = σ(t)R(t)C + a(t)




α ∈ R is the expansion/shrinking rate,

A = −AT ∈ so(n, R) generates the rotations R(t),v is the translation velocity


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X(t) = σ(t)R(t)C + a(t)




α ∈ R is the expansion/shrinking rate,A = −AT ∈ so(n, R) generates the rotations R(t),

v is the translation velocity


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X(t) = σ(t)R(t)C + a(t)




α ∈ R is the expansion/shrinking rate,A = −AT ∈ so(n, R) generates the rotations R(t),v is the translation velocity


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..Self similar solutions without rotationA = 0, α ∈ {0,± 12}, v ∈ Rn

expanding wedge

shrinkers

translating soliton

Abresch-Langer

Circle

Dilation

Expanders X(t) =√

t C, t > 0 Brakke’s 1979 wedgeShrinkers X(t) =

√−t C, t < 0 Circles, Abresch–Langer

curves

Translation: X(t) = C + tv, The Grim Reaper


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..Self similar solutions without rotationA = 0, α ∈ {0,± 12}, v ∈ Rn

expanding wedge

shrinkers

translating soliton

Abresch-Langer

Circle

DilationExpanders X(t) =

√t C, t > 0 Brakke’s 1979 wedge

Shrinkers X(t) =√−t C, t < 0 Circles, Abresch–Langer

curves

Translation: X(t) = C + tv, The Grim Reaper


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What other solitons are there?


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..Non rotating solitons are planar(the case A = 0 is easy)

Theorem. If C is a solution of the non rotating soliton equation

Css = αC + λ(s)Cs + v

then there is a two dimensional affine subspace of Rn that containsthe curve C.

Proof (when v = 0): note that α and λ(s) are scalars. If ϕ, ψ arethe two solutions of the ODE y′′(s) = αy(s) + λ(s)y′(s) with

ϕ(0) = 1, ϕ′(0) = 0, and ψ(0) = 0, ψ′(0) = 1

thenC(s) = ϕ(s)C(0) + ψ(s)Cs(0).

From now on: A ̸= 0!


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..Self similar solutions with rotationwhat kind of rotating solitons can there be?

Given A = −AT ∈ so(n, R), and etA ∈ SO(n, R) we have thefollowing kinds of solitons:

Pure Rotation: X(t) = etACRotation&translation: X(t) = etAC + tv, v ∈ ker ARotation&dilation:

Expanding (t > 0) X(t) =√

t tA CShrinking (t < 0) X(t) = 1√−t (−t)

A C

In both cases, X(t) = (±t)± 12+AC,

(for any z > 0: zA def= e(log z)A)


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Pure Rotation: X(t) = etAC

Rotation&translation: X(t) = etAC + tv, v ∈ ker ARotation&dilation:



A C




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Pure Rotation: X(t) = etACRotation&translation: X(t) = etAC + tv, v ∈ ker A

Rotation&dilation:Expanding (t > 0) X(t) =

√t tA C

Shrinking (t < 0) X(t) = 1√−t (−t)A C




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Pure Rotation: X(t) = etACRotation&translation: X(t) = etAC + tv, v ∈ ker ARotation&dilation:



A C




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..The Soliton Flowthe soliton ODE defines a flow on the unit tangent bundle of Rn

The soliton equation


can be rewritten as a first order system, T = Cs

Cs = T, Ts = pT[(α + A)C + v

]where

pa(b)def= b − ⟨a, b⟩a

is the orthogonal projection along the unit vector a.

The phase space for this system is{(C, T) : ∥T∥ = 1} = Rn × Sn−1.


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.. Is the soliton flow a geodesic flow?

The soliton flow

Cs = T, Ts = pT[(α + A)C + v

]defines a flow on Rn × Sn−1, so through each point C ∈ Rn and ineach direction T ∈ TCRn there is a unique soliton.

Theorem. When A = 0 the soliton flow is the geodesic flow for themetric

(ds)2 = eα2 ∥C∥2∥dC∥2

on Rn.

Huisken’s monotonicity formula:ddt

1√T − t

∫C(t)

e−∥x∥2/4(T−t)ds ≤ 0


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.. Is the soliton flow a geodesic flow?

The soliton flow

Cs = T, Ts = pT[(α + A)C + v

]defines a flow on Rn × Sn−1, so through each point C ∈ Rn and ineach direction T ∈ TCRn there is a unique soliton.Theorem. When A = 0 the soliton flow is the geodesic flow for themetric

(ds)2 = eα2 ∥C∥2∥dC∥2

on Rn.

Huisken’s monotonicity formula:ddt

1√T − t

∫C(t)

e−∥x∥2/4(T−t)ds ≤ 0


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.. Some 3D pictures

In the next series of pictures we have

α ∈ R . . .

α > 0 expandingα < 0 contractingα = 0 no dilation

v =

00vz

, vz ≥ 0,A =

0 −ω 0ω 0 00 0 0

ω > 0


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.. Wagon wheels


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.. Translating and rotating


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From here on assume that v = 0 and α ̸= 0

i.e. only look atrotating shrinkers and expanders


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..The ends of rotating expandersCss = (α + A)C + λCs

Assume α > 0 (expanding solitons)and A ̸= 0 (rotating solitons).

The distance to the origin ∥C∥attains a unique minimum on thesoliton,

∥C∥ → ∞ along both ends of thesolitonAs s → ±∞, we have theasymptotic expansion

C(s) ∼ eθ(s)(α+A)Γ±, θ(s) → ∞

for certain constant vectors Γ±that depend on the soliton.

−100

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The distance to the origin ∥C∥attains a unique minimum on thesoliton,∥C∥ → ∞ along both ends of thesoliton

As s → ±∞, we have theasymptotic expansion

C(s) ∼ eθ(s)(α+A)Γ±, θ(s) → ∞


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The distance to the origin ∥C∥attains a unique minimum on thesoliton,∥C∥ → ∞ along both ends of thesolitonAs s → ±∞, we have theasymptotic expansion

C(s) ∼ eθ(s)(α+A)Γ±, θ(s) → ∞


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Next, rotating shrinkers. . .

α < 0, A =

0 −ω 0ω 0 00 0 0

z-axis

(-α)-½


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.. Why should there be a Lyapunov function?

Gradient flow of∫

L(u, ux)dxEuler

Lagrange−→ ut =∂Lux∂x

− Lu.

E.g.

gradient flow of∫ { 1

2 u2x − F(u)

}dx is ut = uxx + F ′(u).

Traveling waves: u(t, x) = u(x − ct), ut = cuxare solutions of the traveling wave ODE

cux =dLuxdx

− Lu. e.g. cux = uxx + F ′(u).


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..Why should there be a Lyapunov function?Travelling wave ODE: cux =

dLuxdx − Lu

The Hamiltonian H = uxLux − L is a Lyapunov function for thetraveling wave ODE:

ddx

(uxLux − L

)= uxxLux + ux

dLuxdx

− Lu ux − Luxuxx

= ux{dLux

dx− Lu

}= c (ux)2.


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..The almost Lyapunov functionα < 0, v = 0.

If we assume α < 0, then the soliton flow on the unit tangentbundle,

Cs = T, Ts = pT[(α + A)C

]has an almost Lyapunov function, namely

V(C, T) = ⟨T, AC⟩e α2 ∥C∥2 .

One hasdVds

= eα2 ∥C∥2∥pT(AC)∥2.

In particular,dVds

≥ 0 with equality only if AC is a multiple ofT = Cs.

Critical orbits: orbits on which V is constant.


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..The ends of rotating shrinkersV(C, T) = ⟨T, AC⟩eα∥C∥2/2 is an almost Lyapunov function

Let C be a shrinking rotating soliton. If the end {C(s) : s ≥ 0} isbounded, then its ω-limit set is a union of critical orbits. Theonly bounded critical orbit is the circle. So, bounded ends of asoliton converge to the circle.

Some rotating shrinkers have unbounded ends (e.g. the z-axis).

Theorem about unbounded ends. If {C(s) : s ≥ 0} is anunbounded end of a rotating shrinking soliton, then either the end isasymptotic to a logarithmic spiral,

(∃Γ∈Rn) C(s) ∼ e−θ(s)(α+A)Γ, θ(s) → ∞

or else the ω-limit set of C is contained in the z-axis.


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.. Could this happen?

-1

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1-150

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.. This happens. . .


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.. Linearize near the z-axis

zx+iy=f(z)x

y

fzz1 + |fz|2

− zfz + (1 − iω)f = 0.

fzz − zfz + (1 − iω)f = 0.


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.. Linearize near the z-axis

zx+iy=f(z)x

y

fzz − zfz + (1 − iω)f = 0 =⇒ f (z) = aW(z) + bW(−z),

Asymptotics:

W(z) ∼ z1−iω, W(−z) ∼ Kz−2+iωez2/2

|W(z)| ∼ z, |W(−z)| ∼ ez2/2+···


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..Up and down the z-axisf (z) = aW(z) + bW(−z), W(z) ∼ z1−iω , W(−z) ∼ Kz−2+iωez2/2

M1

M2 M3

Grim reapers

Matching the aymptotics leads to the following prediction ofthe growth of the amplitudes Mk of the oscillation:

Mk+1 = e14 M

2k+O(log Mk)


the zoo of curve shortening solitons in rnangenent/preprints/ima-zootalk3.pdf · the zoo of curve...

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