On the Shannon entropy power on Riemannian manifolds and Ricci flow

TL;DR

This paper investigates the properties of Shannon entropy power on Riemannian manifolds and Ricci flows, establishing concavity, convexity, and rigidity results under curvature conditions, with applications to entropy inequalities.

Contribution

It proves new concavity and convexity properties of Shannon entropy power on manifolds and Ricci flows, identifying rigidity models as Einstein, quasi-Einstein, or Ricci solitons.

Findings

Concavity of Shannon entropy power for heat equations on manifolds.

Convexity of Shannon entropy power for conjugate heat equations under Ricci flow.

Entropy isoperimetric inequality on manifolds with non-negative Ricci curvature.

Abstract

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the rigidity models of the Shannon entropy power are Einstein or quasi Einstein manifolds with Hessian solitons. Moreover, we prove the convexity of the Shannon entropy power for the conjugate heat equation introduced by G. Perelman on Ricci flow and that the corresponding rigidity models are the shrinking Ricci solitons. As an application, we prove the entropy isoperimetric inequality on complete Riemannian manifolds with non-negative (Bakry-Emery) Ricci curvature and the maximal volume growth condition.