The logarithmic entropy formula for the linear heat equation on Riemannian manifolds

TL;DR

This paper introduces a new logarithmic entropy functional for the heat equation on Riemannian manifolds, proving its monotonicity under nonnegative Ricci curvature and using it to characterize Euclidean space.

Contribution

It presents a simpler monotonicity result for a logarithmic entropy functional on Riemannian manifolds, extending previous work without Ricci flow.

Findings

The entropy functional is monotone decreasing on manifolds with nonnegative Ricci curvature.

Monotonicity characterizes Euclidean space among Riemannian manifolds.

The approach simplifies previous results by removing the need for Ricci flow.

Abstract

In this paper we introduce a new logarithmic entropy functional for the linear heat equation on complete Riemannian manifolds and prove that it is monotone decreasing on complete Riemannian manifolds with nonnegative Ricci curvature. Our results are simpler version, without Ricci flow, of R.-G. Ye's recent result (arXiv:0708.2008v3). As an application, we apply the monotonicity of the logarithmic entropy functional of heat kernels to characterize Euclidean space.