The Logarithmic Sobolev inequality on non-compact self-shrinkers

TL;DR

This paper proves an optimal logarithmic Sobolev inequality for non-compact self-shrinkers in Euclidean space, extending previous results for closed self-shrinkers and employing the ABP method for the proof.

Contribution

It introduces a sharp logarithmic Sobolev inequality for non-compact self-shrinkers, generalizing prior results and adapting the ABP method for non-compact settings.

Findings

Established an optimal logarithmic Sobolev inequality for non-compact self-shrinkers

Extended Brendle's inequality to non-compact cases

Demonstrated the potential for new inequalities in noncompact manifolds

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.