Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

Brian White

Abstract

For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4π$. In this paper, we use mean curvature flow to show that $k >3π$. We get a slightly larger lower bound for orientable surfaces.

Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

Abstract

For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that . In this paper, we use mean curvature flow to show that . We get a slightly larger lower bound for orientable surfaces.

Paper Structure

This paper contains 12 sections, 25 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. A Bernstein Theorem for Orientable Boundary Shrinkers
  3. A Bernstein Theorem for Orientable Boundary Shrinkers with Singularities
  4. Basic Properties of Entropy and Total Curvature
  5. A general existence theorem for eternal flows
  6. Generalized Möbius Strips
  7. Boundary singularities are unavoidable
  8. The Three Pi Theorem
  9. Oriented Surfaces
  10. Regularity and Compactness
  11. Reducing Total Curvature
  12. Unique Limits

Key Result

Theorem 1

Let $M\subset \mathbf{R}^{n+1}$ be a smoothly embedded, oriented shrinker bounded by an $(n-1)$-dimensional linear subspace $L$. Then $M$ is a half-plane.

Theorems & Definitions (55)

  • Theorem 1
  • proof
  • Claim 1
  • proof : Proof of claim \ref{['maximum-claim']}
  • Remark
  • Theorem 2
  • proof
  • Claim 2
  • Remark 3
  • Definition 4
  • ...and 45 more