W-Entropy, Super Perelman Ricci Flows, and (K, m)-Ricci Solitons

摘要

In this paper, we prove a characterization of (K, infinity)-super Perelman Ricci flows by functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time-dependent metrics and potentials. As a byproduct, we derive the Hamilton type dimension-free Harnack inequality on manifolds with (K, infinity)-super Perelman Ricci flows. Based on a new entropy differential inequality for the heat equation of the Witten Laplacian, we introduce a new W-entropy quantity and prove its monotonicity for the heat equation of the Witten Laplacian On complete Riemannian manifolds with the C D(K, infinity)-condition and on compact manifolds with (K, infinity)-super Perelman Ricci flows. Our results characterize the (K, infinity)-Ricci solitons and the (K, infinity)-Perelman Ricci flows. We also prove an entropy differential inequality on (K, m)-super Perelman Ricci flows, which can be used to characterize the (K, m)-Ricci solitons and the (K, m)-Perelman Ricci flows. Finally, we show that the Gaussian-type solitons on R-m provide asymptotical rigidity models for the W-m,W-K-entropy on manifolds with the CD(-K, m)-condition.