A local regularity theorem for mean curvature flow with triple edges

Felix Schulze; Brian White

doi:10.1515/crelle-2017-0044

Published by De Gruyter November 12, 2017

A local regularity theorem for mean curvature flow with triple edges

Felix Schulze and Brian White

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2017-0044

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Abstract

Mean curvature flow of clusters of n-dimensional surfaces in ℝn+k that meet in triples at equal angles along smooth edges and higher order junctions on lower-dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1440140

Award Identifier / Grant number: DMS-1404282

Funding statement: The research of the first author was partially supported by NSF grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2016 semester. The research of the second author was partially supported by NSF grant DMS-1404282.

A Initial regularity

We verify that any local solution to Brakke flow, which can be written as the graph of a C1-function and has smooth initial data, is actually locally smooth and attains its initial data smoothly. We work again with parabolic Cq,α-spaces.

Theorem A.1.

Let U:=B1⁢(0)⊂Rn and assume that u:U×[0,1)→Rk is C1,α and that graph⁡(u) defines a unit density Brakke flow. If u⁢(⋅,0):U→Rk is smooth, then u:U×[0,b)→Rk is smooth.

Proof.

Let the nonparametric equation for mean curvature flow be given by

∂t⁡u=Q⁢(u,D⁢u,D⁢u2).

We extend u to U×(-1,1) by setting

u⁢(x,t)=u⁢(x,0)+t⁢Q⁢(u⁢(x,0),D⁢u⁢(x,0),D2⁢u⁢(x,0))

for t<0. Note that since u⁢(⋅,0) is smooth, u is smooth on U×(-1,0] and C1,α on U×(-1,1). The graph of u⁢(⋅,t) does not anymore constitute a Brakke flow for t∈(-1,1), but it does so with transport term as follows: For x∈ℝn+k write x=(x′,x′′) where x′∈ℝn,x′′∈ℝk and let f:U×(-1,1)→ℝk be defined by

f⁢(x,t)=Q⁢(u⁢(x′,0),D⁢u⁢(x′,0),D2⁢u⁢(x′,0))-Q⁢(u⁢(x′,t),D⁢u⁢(x′,t),D2⁢u⁢(x′,t))

for t≤0 and f⁢(x,t)=0 otherwise. Note that f is C0,α on U×(-1,1). Then graph⁡(u) constitutes a Brakke flow with transport term as considered in [18]. We can thus apply [18, Theorem 6.3] (with g=0) to see that u is C2,α on U×(-1,1). Therefore u is C2,α on U×[0,1). Now by standard parabolic PDE theory, u is smooth on U×[0,1). ∎

Remark A.2.

The same proof shows that if the assumption u⁢(⋅,0) is smooth is relaxed to u⁢(⋅,0) is in Cq,α for q≥4, then u is Cq,α on U×[0,1).

Corollary A.3.

Let U:=B1⁢(0)⊂Rn and assume that

u:U×[0,1)→ℝk

is C1⁢(U×[0,1))∩C∞⁢(U×(0,1)) and that graph⁡(u) defines a unit density Brakke flow. If u⁢(⋅,0):U→Rk is smooth, then u:U×[0,b)→Rk is smooth.

Proof.

To apply Theorem A.1, we need to show that u is C1,α on U×[0,1). First note that for higher codimension mean curvature flow, balls of radius R⁢(t)=(R0-2⁢n⁢t) still act as barriers. Thus we can adapt the final barrier argument in [5, Section 9.11] to show that u∈C1,1⁢(U×[0,1)). ∎

Remark A.4.

It suffices to assume that u∈C1⁢(U×[0,1)) since Brakke’s local regularity theorem [2] (or alternatively [18]) implies that u is smooth for t>0. By the proof of Proposition 5.3, this even extends to unit regular Brakke flows: the initial surface is attained locally in C1 and thus the higher initial regularity extends.

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Received: 2017-06-09

Revised: 2017-08-24

Published Online: 2017-11-12

Published in Print: 2020-01-01

A local regularity theorem for mean curvature flow with triple edges

Abstract

A Initial regularity

Theorem A.1.

Proof.

Remark A.2.

Corollary A.3.

Proof.

Remark A.4.

References

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