The logarithmic Sobolev inequality for a submanifold in manifold with nonnegative sectional curvature

TL;DR

This paper establishes a sharp logarithmic Sobolev inequality for compact submanifolds within Riemannian manifolds of nonnegative sectional curvature, extending Euclidean results to more general ambient spaces.

Contribution

It generalizes the logarithmic Sobolev inequality to submanifolds in manifolds with nonnegative sectional curvature, including mean curvature terms, extending Brendle's Euclidean results.

Findings

Proves a sharp logarithmic Sobolev inequality for submanifolds in curved ambient spaces.

Includes mean curvature in the inequality, similar to Michael-Simon Sobolev inequality.

Extends previous Euclidean results to more general Riemannian manifolds.

Abstract

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.