Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

Or Hershkovits, Brian White

Abstract

Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology. We show that the entropy of $M$ is greater than or equal to the entropy of a round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in ${\mathbf R}^{k+1}$.

Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

Abstract

Let be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial homology. We show that the entropy of is greater than or equal to the entropy of a round -sphere, and that if equality holds, then is a round -sphere in .

Paper Structure

This paper contains 3 sections, 5 theorems, 25 equations.

Table of Contents

  1. introduction
  2. Proof of the Sharp Entropy Bounds
  3. Motion by Mean Curvature Plus an Ambient Vectorfield

Key Result

Theorem 1

Suppose that $M\subset \mathbf{R}^{m+1}$ is a codimension-one, smooth, closed self-shrinker with nontrivial $k^{\rm th}$ homology. Then the entropy of $M$ is greater than or equal to the entropy of a round $k$-sphere. If equality holds, then $M$ is a round $k$-sphere in $\mathbf{R}^{k+1}$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['intro-main-theorem']}
  • proof : Proof of Theorem \ref{['main-homotopy']}
  • Definition 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Theorem 7: Clearing Out Theorem
  • proof