On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds

TL;DR

This paper extends the understanding of Renyi entropy power's concavity and related inequalities to Riemannian manifolds with non-negative Ricci curvature, revealing geometric conditions for entropy behavior and establishing new inequalities.

Contribution

It proves the concavity of Renyi entropy power on Riemannian manifolds, characterizes rigidity models, and establishes entropy inequalities extending Euclidean results.

Findings

Concavity of Renyi entropy power on manifolds with non-negative Ricci curvature

Rigidity models are Einstein or quasi-Einstein manifolds

Entropy isoperimetric and Gagliardo-Nirenberg-Sobolev inequalities established

Abstract

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savar\'e and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special $(K,m)$-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani \cite{LNVV}, we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with $CD(0,m)$-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power $N_p$, the $p$-th Fisher information $I_p$ and the time derivative of the $W$-entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or $CD(0, m)$-condition and maximal volume growth condition.