Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mean Curvature Flow with Boundary

Brian White

Abstract

We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.

Mean Curvature Flow with Boundary

Abstract

We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.

Paper Structure

This paper contains 20 sections, 42 theorems, 228 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. $L^p$ vectorfields
  4. A Varifold Closure Theorem
  5. Brakke Flows with Boundary
  6. Monotonicity with Boundary in a Manifold
  7. Monotonicity with Boundary in Euclidean Space
  8. Entropy
  9. Entropy and Maximal Density Ratio
  10. Compactness Theorems
  11. Tangent Flows
  12. Mod $2$ Flat Chains
  13. Standard Brakke Flows and The Closure Theorem
  14. Existence
  15. Self-Similar Flows
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $N$ be a smooth, compact, $(m+1)$-dimensional Riemannian manifold with smooth, strictly mean-convex boundary. Let $M_0$ be a smoothly embedded $m$-dimensional submanifold of $N$ whose boundary is a smooth submanifold $\Gamma$ of $\partial N$. (More generally, $M_0$ can be any $m$-rectifiable set with boundary $\Gamma$ such that $M(0)=M_0$. Furthermore, if $M(\cdot)$ is any standard Brakke flow

Figures (1)

  • Figure :

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • Definition 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 76 more