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The Logarithmic Sobolev inequality on non-compact self-shrinkers

Guofang Wang, Chao Xia, Xiqiang Zhang

Abstract

In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds.

The Logarithmic Sobolev inequality on non-compact self-shrinkers

Abstract

In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds.

Paper Structure

This paper contains 3 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. A proof of the classical LSI
  3. LSI for noncompact self-schrinkers

Key Result

Theorem 1.1

Let $\Sigma$ be an $n$-dimensional complete, non-compact properly embedded self-shrinker in $\mathbb{R}^{n+m}$. Then for any positive function $\varphi$ on $\Sigma$ with $\int_\Sigma \varphi \,d \gamma =1$, there holds Equivalently, for any positive function $f$ on $\Sigma$ with $\int_\Sigma f dvol =1$, there holds

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 18 more