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Isoperimetry by stretching

Kobe Marshall-Stevens, Gongping Niu

Abstract

We construct isoperimetric regions from separating hypersurfaces in closed manifolds. This yields isoperimetric boundaries exhibiting a wide variety of topological types and singular sets.

Isoperimetry by stretching

Abstract

We construct isoperimetric regions from separating hypersurfaces in closed manifolds. This yields isoperimetric boundaries exhibiting a wide variety of topological types and singular sets.
Paper Structure (7 sections, 5 theorems, 16 equations, 1 figure)

This paper contains 7 sections, 5 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strategy
  3. Notation
  4. Stretching
  5. Applications
  6. Topological types
  7. Singular sets

Key Result

Theorem 1

If $\Sigma \subset M$ is a smooth, closed, connected, separating hypersurface which is uniquely homologically area-minimising in a tubular neighbourhood of $\Sigma$ in $(M,g)$, then there is a Riemannian metric, $h$, on $M$ such that $\Sigma$ bounds an isoperimetric region in $(M,h)$.

Figures (1)

  • Figure 1: In each of the two graphics above, the three circles depict $\Sigma$ between the boundaries of the tubular neighbourhood in which it is uniquely area-minimising. Along the stretching procedure, the 'caps' outside of this tubular neighbourhood are unchanged.

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Corollary 2
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • ...and 8 more