Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces

Abstract

By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the W-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

30 pages

References in corpus (3)

• Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow
• Rigidity of gradient Ricci Solitons
• $W$-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds

Cited by in corpus (2)

• On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds
• On the Shannon entropy power on Riemannian manifolds and Ricci flow