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Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

Vedansh Arya, Seongmin Jeon, Vesa Julin

Abstract

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.

Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

Abstract

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.
Paper Structure (17 sections, 19 theorems, 278 equations)

This paper contains 17 sections, 19 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Set-up and statement of the flatness decay
  4. The flat flow solution
  5. Flatness decay
  6. The sub- and supergraphs
  7. Preliminary results
  8. Weak Harnack inequality
  9. The ABP-estimate
  10. Basic measure estimate
  11. Weak Harnack inequality
  12. Flatness decay
  13. Oscillation decay
  14. Flatness decay
  15. Final regularity estimate and the proof of the main theorem
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\{E(t)\}_{t\geq 0}$ be a volume-preserving flat flow in $\mathbb{R}^{n+1}$, for $n\leq 2$, starting from a bounded set of finite perimeter $E_0 \subset\mathbb{R}^{n+1}$ with volume $|E_0| = |B_r|$ and assume Then there exists a time $T_0 >0$ such that the sets $E(t)$ are smooth for all $t \in (T_0,\infty)$, they solve eq:VMCF in the classical sense and converge exponentially fast to $B_r(x_0

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 28 more