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In this paper, we prove the concavity of the Shannon entropy power for theheat equation associated with the Laplacian or the Witten Laplacian on completeRiemannian manifolds with suitable curvature-dimension condition and on compactsuper Ricci flows. Under suitable curvature-dimension condition, we prove thatthe rigidity models of the Shannon entropy power are Einstein or quasi Einsteinmanifolds with Hessian solitons. Moreover, we prove the convexity of theShannon entropy power for the conjugate heat equation introduced by G. Perelmanon Ricci flow and that the corresponding rigidity models are the shrinkingRicci solitons. As an application, we prove the entropy isoperimetricinequality on complete Riemannian manifolds with non-negative (Bakry-Emery)Ricci curvature and the maximal volume growth condition.

On the Shannon entropy power on Riemannian manifolds and Ricci flow