On the $W$-entropy and Shannon entropy power on RCD$(K,N)$ and RCD$(K,n,N)$ spaces

Abstract

In this paper, we prove the $W$-entropy formula and the monotonicity and rigidity theorem of the $W$-entropy for the heat flow on RCD$(K,N)$ and RCD$(K,n,N)$ spaces $(X,d,\mu )$, where $K\in \mathbb{R}$, $n\in \mathbb{N}$ is the geometric dimension of $(X,d,\mu )$ and $N\ge n$. We also prove the $K$-concavity of the Shannon entropy power on RCD$(K,N)$ spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam type logarithmic Sobolev inequality on RCD$(0,N)$ spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam type logarithmic Sobolev inequality with sharp constant on noncollapsing RCD$(0,N)$ spaces.

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