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The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

Lei Liu, Guofang Wang, Liangjun Weng

Abstract

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore, we use this estimate and the idea of Brendle in his recent work \cite{Brendle2019} to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in $\mathbb{R}^{n+m}$, which is optimal and gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6, when the codimension $m\leq 2$.

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

Abstract

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore, we use this estimate and the idea of Brendle in his recent work \cite{Brendle2019} to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in , which is optimal and gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6, when the codimension .

Paper Structure

This paper contains 5 sections, 15 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. A generalized cone
  3. A new proof of the relative isoperimetric inequality
  4. relative isoperimetric inequality for minimal submanifolds
  5. Appendix

Key Result

Theorem 1.1

Let $M\subset\mathbb{R}^{n+m}$$(m\ge 1)$ be a compact $n$-dimensional submanifold with boundary $\partial M$, then where $b_{n,m}$ is defined by with equality for $m\le 2$ if and only if $M$ is a round ball. Here $H$ is the mean curvature of $M$, $|\partial M|$ and $|M|$ are the area and the volume of $\partial M$ and $M$ respectively.

Theorems & Definitions (27)

  • Theorem 1.1: Brendle Brendle2019
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Choe-Ghomi-Ritoré CGR07
  • Proposition 2.1: Choe-Ghomi-Ritoré CGR06
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Remark 2.5
  • ...and 17 more