Tohoku Mathematical Journal
2024
December
SECOND SERIES VOL. 76, NO. 4

Tohoku Math. J.
76 (2024), 577-608

Title ON THE SHANNON ENTROPY POWER ON RIEMANNIAN MANIFOLDS AND RICCI FLOW

Dedicated to the memory of the second author's mother
Author Songzi Li and Xiang-Dong Li

(Received March 4, 2021, revised September 11, 2023)
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.

Mathematics Subject Classification. Primary 53C44; Secondary 58J35, 58J65.
Key words and phrases. Boltzmann-Shannon entropy, Einstein manifolds, Hessian solitons, Ricci flow, Shannon entropy power.

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