On the Shannon entropy power on Riemannian manifolds and Ricci flow

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.

A generalization of Costa's Entropy Power Inequality

“…as recently proved in [15]. In a completely analogous fashion, if we move from the Euclidean to the Riemannian framework, Theorem 1.1 reads as follows.…”

On the p-Renyi entropy power for nonlinear diffusion equations on Riemannian manifolds

“…On the other hand, S. Li and X.-D Li [3,4] proved the concavity of Shannon entropy power for the Witten Laplacian heat equation ∂ t u = Lu := ∆u − ∇φ • ∇u and the concavity of Rényi entropy power for the porous medium equation ∂ t u = Lu γ with γ > 1 on weighted Riemannian manifolds with CD(0, m) or CD(−K, m) conditions, and also on (0, m) or (−K, m) super Ricci flows.…”

“…Motivated by the works of [8,6,3,4], in [11] and [10], the first author and coauthors studied the concavities of the p-Shannon entropy power and the p-Rényi entropy power for positive solutions to the p-heat equations ∂ t u p−1 = (p − 1) p−1 ∆ p u and the doubly nonlinear diffusion equations on Riemannian manifolds with nonnegative Ricci curvature, respectively. More precisely, let u be a positive solution to the doubly nonlinear diffusion equation…”

Curvature-dimension conditions for symmetric quantum Markov semigroups

“…In [Vil00] Villani gave a short proof and remarked that this can be proved using Γ 2 -calculus. Recently Li and Li [LL20] considered this problem on the Riemannian manifold under the curvature-dimension condition CD(K, N). Here we show that the geodesic concavity of the entropy power follows from the (K, N)-convexity of the entropy.…”