Year: 2024
Type: article
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 NFW formula which indicates the intrinsic relationship between the Shannon entropy power $\mathcal{N}$, Perelman's $\mathcal{F}$-functional and $\mathcal{W}$-entropy on the Ricci flow. This leads us to prove that the Shannon entropy power for the conjugate heat equation on Ricci flow is convex and 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 ($m$-dimensional Bakry-Emery) Ricci curvature and the maximal volume growth condition.
Source: Tohoku Mathematical Journal
DOI: 10.2748/tmj.20230403
Authors: Songzi Li, Xiang-Dong Li
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: 最优传输问题与流体力学方程的概率研究
OpenAlex ID: W4287907905