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

In this paper, we prove a characterization of \((K, \infty )\)-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 \((K, \infty )\)-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 \(CD(K, \infty )\)-condition and on compact manifolds with \((K, \infty )\)-super Perelman Ricci flows. Our results characterize the \((K, \infty )\)-Ricci solitons and the \((K, \infty )\)-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 \(\mathbb {R}^m\) provide asymptotical rigidity models for the \(W_{m, K}\)-entropy on manifolds with the \(CD(-K, m)\)-condition.