Dynamics of convex mean curvature flow

Sigurd Angenent; Panagiota Daskalopoulos; Nataša Šešum

doi:10.1515/crelle-2025-0029

Article

Dynamics of convex mean curvature flow

Sigurd Angenent , Panagiota Daskalopoulos and Nataša Šešum

Published/Copyright: May 3, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 825

Abstract

There is an extensive and growing body of work analyzing convex ancient solutions to mean curvature flow (MCF), or equivalently of rescaled mean curvature flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space 𝑋 of all convex hypersurfaces M ⊂ R n + 1 , regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well-known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary noncompact and nonsmooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space 𝑋 of convex hypersurfaces and show that RMCF defines a continuous local semiflow on 𝑋 whose fixed points are the shrinking cylinder solitons S k × R n − k , and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on 𝑋. We consider the set of all hypersurfaces that lie on an ancient solution that, in backward time, is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of 𝑋, and considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Čech cohomology. We also prove that the space of all convex ancient solutions with point symmetry in R n + 1 is homeomorphic to an ( n − 1 ) -dimensional simplex, in the case when n = 2 or n = 3 , and conjecture that it holds true for any n ≥ 2 .

A The structure of complete convex hypersurfaces

In this appendix, we present some basic known results on the general structure of convex hypersurfaces. Convex hypersurfaces are studied in both convex geometry and differential geometry of submanifolds. As a result, there are different notions of convexity preferred in different subjects. In this paper, we use the following definition.

Definition A.1

Definition A.1 (Convex hypersurfaces in R n + 1 )

A hypersurface M ⊂ R n + 1 is called convex if it is the boundary of a convex set 𝐶 of nonempty interior.

The following is a known observation on the structure of convex sets. For its proof, we refer the reader to [13]. However, this result has been known for a long time (see for example the work by Hung-Hsi Wu [41]).

Proposition A.2

Let M = ∂ C be the boundary of a closed convex set C ⊂ R n + 1 with nonempty interior, that is, 𝑀 is a convex hypersurface in R n + 1 . Then either M = R n or M = R k × M ̂ for some 0 ≤ k < n and M ̂ = ∂ C ̂ , where C ̂ ⊂ R n + 1 − k is a closed convex set with nonempty interior which contains no infinite line. Moreover, such M ̂ is either homeomorphic to S n − k or R n − k . In the former case, M ̂ = ∂ C ̂ is a compact hypersurface.

The following theorem shown in [41] concerns with complete noncompact hypersurfaces 𝑀 that contain no infinite line.

Proposition A.3

Proposition A.3 (Hung-Hsi Wu [41])

Assume that M = ∂ C is a convex hypersurface in R n + 1 that is homeomorphic to R n and contains no infinite line. Then coordinates can be so chosen that { x n + 1 = 0 } is a supporting hyperplane to the convex set 𝐶 at the origin, and it has the following additional properties.

Let D : = π ( C ) , where π : R n + 1 → { x n + 1 = 0 } is the standard orthogonal projection, and let D ∘ be its interior relative to the hyperplane { x n + 1 = 0 } . Then 𝑀 can be expressed as the graph of a nonnegative convex function u : D ∘ → R . If 𝑀 is C ∞ , then 𝑢 is a C ∞ function on D ∘ .
For every x 0 ∈ D ∖ D ∘ , π − 1 ⁢ ( x 0 ) is a semi-infinite line segment.
If the image γ ⁢ ( M ) of the Gauss map γ : M → S n has nonempty interior relative to S n , then for any c > 0 , the level set M ∩ { x n + 1 = c } is homeomorphic to S n − 1 . This homeomorphism is a diffeomorphism if 𝑀 is C ∞ smooth.

Since the set D : = π ( C ) that is defined in the previous proposition will play crucial role in the discussion of uniqueness of MCF solutions, we will refer to it as the shadow of 𝑀 or 𝐶.

Definition A.4

Definition A.4 (The shadow of a convex hypersurface)

Let M = ∂ C be a convex hypersurface in R n + 1 that is homeomorphic to R n and contains no infinite line. The set D : = π ( C ) , defined as in Proposition A.3, is called the shadow of 𝑀 or 𝐶.

Below, we summarize more classical facts about convex sets. Let C ⊂ R n + 1 be a closed convex set.

Lemma A.5

Lemma A.5 (When 𝐶 contains a line)

If 𝐶 contains a line ℓ ⊂ C and x ∈ C is given, then 𝐶 contains the line through 𝑥 that is parallel to ℓ. In particular, 𝐶 is a product of ℓ and a convex set C ′ ⊂ ℓ ⟂ .

Proof

Assume ℓ = { a + σ ⁢ b ∣ σ ∈ R } for suitable vectors a , b ∈ R n + 1 and let x ∈ C be given. We will show { x + σ ⁢ b ∣ σ ∈ R } ⊂ C .

For any ρ ∈ R , the line segment connecting 𝑥 and a + ρ ⁢ b is contained in 𝐶. Thus, for all ρ ∈ R and θ ∈ [ 0 , 1 ] ,

( 1 − θ ) ⁢ x + θ ⁢ a + θ ⁢ ρ ⁢ b ∈ C .

For n = 1 , 2 , 3 , … , choose θ n = n − 1 , ρ n = n ⁢ σ . Then

x n = ( 1 − n − 1 ) ⁢ x + n − 1 ⁢ a + σ ⁢ b ∈ C .

Let n → ∞ and recall that 𝐶 is closed to conclude that 𝐶 contains lim n → ∞ x n = x + σ ⁢ b . ∎

Lemma A.6

If 𝐶 contains a ray ℓ + = { a + σ ⁢ b ∣ σ ⩾ 0 } and if x ∈ C is any point, then 𝐶 contains the ray starting at 𝑥 in the same direction as ℓ + , i.e. { x + σ ⁢ b ∣ σ ≥ 0 } .

Proof

The proof is the same as in the previous case. ∎

Lemma A.7

If the closed convex set C ⊂ R n + 1 does not contain a ray, then 𝐶 is compact.

Proof

Let a ∈ C and, assuming 𝐶 is not bounded, we choose a sequence of points p k ∈ C with ∥ p k ∥ → ∞ . The line segments connecting 𝑎 and p k are all contained in 𝐶. Let b k = p k / ∥ p k ∥ ∈ S n , and pass to a subsequence for which b k → B ∈ S n . Then the points x k = a + σ ⁢ b k all belong to 𝐶 provided 0 ≤ σ ≤ ∥ p k ∥ . We let k → ∞ and find that

lim x k = a + σ ⁢ b ∈ C for any ⁢ σ ≥ 0 .

Hence 𝐶 contains the ray in the direction 𝑏 starting at 𝑎. ∎

Lemma A.8

If h : B R ⁢ ( 0 ) → R is a convex function that is bounded by | h ⁢ ( x ) | ≤ M for all x ∈ B R ⁢ ( 0 ) , then ℎ is Lipschitz on B ( 1 − δ ) ⁢ R ⁢ ( 0 ) with Lipschitz constant 2 ⁢ M / δ ⁢ R .

Proof

Let p , q ∈ B ( 1 − δ ) ⁢ R ⁢ ( 0 ) with h ⁢ ( q ) > h ⁢ ( p ) be given, and extend the line segment p ⁢ q until it intersects ∂ B R ⁢ ( 0 ) , say at the point 𝑟. Then the length of q ⁢ r is at least δ ⁢ R . Convexity of ℎ implies

h ⁢ ( q ) − h ⁢ ( p ) ∥ q − p ∥ ≤ h ⁢ ( r ) − h ⁢ ( q ) ∥ r − q ∥ ≤ 2 ⁢ M δ ⁢ R . ∎

This directly implies the following.

Lemma A.9

Let C ⊂ R n + 1 be a closed convex set, and let p , q be two points with B δ ⁢ ( p ) ⊂ int C and B δ ⁢ ( q ) ∩ C = ∅ . Let p ⁢ q be the line segment connecting 𝑝 and 𝑞. Then ∂ C ∩ B δ / 2 ⁢ ( p ⁢ q ) is the graph of a Lipschitz continuous function in a coordinate system in which the segment p ⁢ q is the vertical axis. The Lipschitz constant of the function is bounded by 2 ⁢ ∥ p − q ∥ / δ .

B Some useful facts

A family of hypersurfaces M t ⊂ R n × R of the form

M t = { ( x , y ) ∈ R n × R ∣ ∥ x ∥ = r ⁢ ( y , t ) }

evolves by MCF if

r t = r y ⁢ y 1 + r y 2 − n − 1 r .

B.1 Expanding solitons from cones

The hypersurfaces M t are a self-similar expanding soliton exactly when r ⁢ ( y , t ) = t ⁢ E ⁢ ( y / t ) for some function E : R → ( 0 , ∞ ) , which then must satisfy

(B.1) E ′′ ⁢ ( η ) 1 + E ′ ⁢ ( η ) 2 + η 2 ⁢ E ′ ⁢ ( η ) − 1 2 ⁢ E ⁢ ( η ) − n − 1 E ⁢ ( η ) = 0 .

We recall that it was shown in [6, 27] that, for each a > 0 , there is a unique solution E a : R → R of (B.1) with E a ⁢ ( 0 ) = a , E a ′ ⁢ ( 0 ) = 0 . This solution is an even function which is defined for all η ∈ R , which is strictly increasing for η > 0 , and for which

lim η → ∞ E a ⁢ ( η ) η = A a > 0

exists.

Let R > 0 be given. Let u ∈ R n × { 0 } be any unit vector, and consider the solution to MCF given by

( x ̂ , y ̂ ) ∈ M t ⇔ ∥ x ̂ − R ⁢ u ∥ = t ⁢ E 2 ⁢ R ⁢ ( y ̂ t ) for all ⁢ ( x ̂ , y ̂ ) ∈ R n × R .

For t ↘ 0 , this solution converges to the cone ∥ x ̂ − R ⁢ u ∥ = A ⁢ | y ̂ | , where

A = A 2 ⁢ R = lim η → ∞ E 2 ⁢ R ⁢ ( η ) / η .

We get a solution to RMCF by setting x ̂ = e − τ / 2 ⁢ x , y ̂ = e − τ / 2 ⁢ y , and t = 1 − e − τ as follows:

∥ x − R ⁢ e τ / 2 ⁢ u ∥ = e τ − 1 ⁢ E 2 ⁢ R ⁢ ( y e τ − 1 ) .

Lemma B.1

For any R > 0 , there is an A > 0 such that, for any C ∈ X which lies in the region { ( x , y ) ∈ R n × R ∣ ∥ x − R ⁢ u ∥ ≥ A ⁢ | y | } , the set ϕ τ ⁢ ( C ) lies in the region ∥ x ∥ ≥ R if τ ≥ 2 .

Proof

Given 𝑅, we choose A = A 2 ⁢ R as above. If 𝐶 is a closed convex set that lies in the region ∥ x − R ⁢ u ∥ ≥ A ⁢ | y | , then ϕ τ ⁢ ( C ) is contained in the region

∥ x − R ⁢ e τ / 2 ⁢ u ∥ ≥ E 2 ⁢ R ⁢ ( 0 ) ⁢ e τ − 1 = 2 ⁢ R ⁢ e τ − 1 .

This implies

∥ x ∥ ≥ ∥ x − R ⁢ e τ / 2 ⁢ u ∥ − R ⁢ e τ / 2 ≥ ( 2 ⁢ e τ − 1 − e τ / 2 ) ⁢ R .

If τ ≥ 2 , then 2 ⁢ e τ − 1 − e τ / 2 ≥ 2 ⁢ e 2 − 1 − e ≈ 2.337 ⁢ … > 1 . ∎

B.2 The BLT pancake

Bourni, Langford, and Tinaglia [10] established the existence and uniqueness of an O ⁢ ( n ) × O ⁢ ( 1 ) symmetric ancient solution that fills the slab R n × ( − π , π ) as t → − ∞ .

We will denote the BLT pancake of width 𝜋 that becomes singular at time t = 0 by P ̂ t ⊂ R n + 1 , t < 0 . For t → − ∞ , the pancake P t is contained in a pill-box B R ⁢ ( t ) n × [ − π , π ] whose radius is asymptotically given by R BLT ⁢ ( t ) = − t + o ⁢ ( t ) . For t ↗ 0 , the pancake shrinks to a round point at the origin, according to Huisken and Gage–Hamilton’s theorem.

Lemma B.2

There is a ϱ > 0 such that, for any δ > 0 , there exist r , T > 0 with the property that, for any C ∈ X that is disjoint from the set B r n × [ − δ , δ ] ⊂ R n + 1 , the image ϕ T ⁢ ( C ) is disjoint from B ϱ / δ n + 1 .

Proof

For any θ ∈ R , we consider the time translate P ̂ t − 2 which becomes singular at time t = 2 . The corresponding RMCF is given by P τ = e τ / 2 ⁢ P ̂ − 1 − e − τ . Next we translate in the rescaled time by an amount ϑ > 0 and consider

Q τ : = P τ − ϑ = e ( τ − ϑ ) / 2 P ̂ − 1 − e ϑ − τ .

At time τ = 0 , we have Q 0 = e − ϑ / 2 ⁢ P ̂ − 1 − e ϑ , which is contained in

B R BLT ⁢ ( − 1 − e ϑ ) n × [ − π ⁢ e − ϑ / 2 , π ⁢ e − ϑ / 2 ] .

We choose ϑ = 2 ⁢ ln ⁡ π δ , so that π ⁢ e − ϑ / 2 = δ , and let

r = R BLT ⁢ ( − 1 − e ϑ ) = R BLT ⁢ ( − 1 − π 2 / δ 2 ) .

This ensures that Q 0 ⊂ B r n × [ − δ , δ ] so that Q 0 is disjoint from 𝐶.

Choose T = 2 ⁢ ϑ . Then ϕ T ⁢ ( C ) must be disjoint from

Q T = e ϑ / 2 ⁢ P ̂ − 1 − e − ϑ = π δ ⁢ P ̂ − 1 − δ 2 / π 2 .

We may assume that δ < π , so that 1 < 1 + δ 2 / π 2 < 2 . Let ϱ > 0 be small enough so that B ϱ / π n + 1 is contained in P ̂ t for all t ∈ [ − 2 , − 1 ] . These choices imply that ϕ T ⁢ ( C ) is disjoint from π δ ⁢ B ϱ / π n + 1 = B ϱ / δ n + 1 , as claimed. ∎

B.3 Decay of the Gaussian mass

Even though we did not use it in this paper, the following result does seem like it might be useful in another setting.

Lemma B.3

On any time interval [ τ 0 , τ 1 ] , the mass of C τ is bounded from below by

m ⁢ ( C τ 1 ) ≥ m ⁢ ( C τ 0 ) − 1 2 ⁢ H ⁢ ( C τ 0 ) ⁢ τ 1 − τ 0 .

Proof

The normal velocity of ∂ C τ is V = H + 1 2 ⁢ ⟨ x , N ⟩ , so the rate at which the mass of C τ decreases satisfies

− d ⁢ m ⁢ ( C τ ) d ⁢ τ = ∫ ∂ C τ e − ∥ x ∥ 2 / 4 ⁢ V ⁢ d ⁢ H n ( 4 ⁢ π ) n / 2 ≤ ( ∫ ∂ C τ e − ∥ x ∥ 2 / 4 ⁢ d ⁢ H n ( 4 ⁢ π ) n / 2 ) 1 / 2 ⁢ ( ∫ ∂ C τ e − ∥ x ∥ 2 / 4 ⁢ V 2 ⁢ d ⁢ H n ( 4 ⁢ π ) n / 2 ) 1 / 2 = H ⁢ ( C τ ) ⁢ ( ∫ ∂ C τ e − ∥ x ∥ 2 / 4 ⁢ V 2 ⁢ d ⁢ H n ( 4 ⁢ π ) n / 2 ) 1 / 2 .

The last integral is the rate at which H ⁢ ( C τ ) decreases,

∫ ∂ C τ e − ∥ x ∥ 2 / 4 ⁢ V 2 ⁢ d ⁢ H n ( 4 ⁢ π ) n / 2 = − d ⁢ H ⁢ ( C τ ) d ⁢ τ .

We therefore get

− d ⁢ m ⁢ ( C τ ) d ⁢ τ ≤ − H ⁢ ( C τ ) ⁢ d ⁢ H ⁢ ( C τ ) d ⁢ τ = − 1 2 ⁢ d ⁢ H ⁢ ( C τ ) 2 d ⁢ τ .

On any time interval [ τ 0 , τ 1 ] , we therefore have

m ⁢ ( C τ 0 ) − m ⁢ ( C τ 1 ) ≤ ∫ τ 0 τ 1 − 1 2 ⁢ d ⁢ H ⁢ ( C τ ) 2 d ⁢ τ ⁢ d τ ≤ ∫ τ 0 τ 1 1 2 ⁢ d τ ⁢ − ∫ τ 0 τ 1 d ⁢ H ⁢ ( C τ ) 2 d ⁢ τ ⁢ d τ ≤ 1 2 ⁢ ( τ 1 − τ 0 ) ⁢ H ⁢ ( C τ 0 ) 2 − H ⁢ ( C τ 1 ) 2 ≤ 1 2 ⁢ H ⁢ ( C τ 0 ) ⁢ τ 1 − τ 0 . ∎

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Received: 2023-07-25

Revised: 2025-03-06

Published Online: 2025-05-03

Published in Print: 2025-08-01

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