Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows

Or Hershkovits, Brian White

Abstract

We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (Sept 15, 2023) corrects a minor typo.

Avoidance for Set-Theoretic Solutions of Mean-Curvature-Type Flows

Abstract

We provide a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint as long as one of the subsolutions is compact; previously, this was only known for Euclidean space (with no ambient vectorfield). The new version (Sept 15, 2023) corrects a minor typo.

Paper Structure

This paper contains 13 sections, 28 theorems, 163 equations.

Table of Contents

  1. introduction
  2. Basic Definitions
  3. Elementary properties of weak set flows
  4. Barrier Modification
  5. Bounds on the Distance Function
  6. The Avoidance Theorem
  7. Equivalent Definitions of Weak Set Flow
  8. The Biggest flow
  9. Limits of Weak Set Flows
  10. Boundaries
  11. Mean curvature flow with a transport term
  12. $X$-mean-convex flows
  13. Varifold flows

Key Result

Theorem 3

Let $m=\dim N -1$ and $c>2m$. Given $p\in N$, there exists an $\epsilon>0$ with the following property.

Theorems & Definitions (61)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6: Ilmanen ilmanen-generalized*Theorem 6.4
  • proof
  • ...and 51 more