On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds (2001.11184v1)

Abstract: In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savar\'e and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special $(K,m)$-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani \cite{LNVV}, we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with $CD(0,m)$-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power $N_p$, the $p$-th Fisher information $I_p$ and the time derivative of the $W$-entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or $CD(0, m)$-condition and maximal volume growth condition.