The W-entropy is introduced by G. Perelman in his seminal work on Ricci flow, and this notion is brought to (time-homogeneous) Riemannian manifolds by L. Ni. He proves that, on Riemannian manifolds with nonnegative Ricci curvature, the W-entropy is monotone in time along the heat flow. Moreover, this monotonicity enjoys a rigidity in the sense that vanishing time derivative can happens only in a very special case. Indeed, the space must be Euclidean.

In this talk, we show the corresponding results on Riemannian metric measure spaces with nonnegative Ricci curvature and upper dimension bound (i.e., RCD (0,N) spaces). Because of the lack of usual differentiable structure, we have to develop new approaches. As a by-product, some of our results are new even on Riemannian manifolds. In addition, we can find more spaces than Euclidean spaces in the class of RCD (0,N) spaces in the rigidity.

This talk is based on a joint work with X.-D. Li (Chinese Academy of Science).