On the Shannon entropy power on Riemannian manifolds and Ricci flow

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.