$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons

In this paper, we first prove the equivalence between the $(K, \infty)$-super Perelman Ricci flows and two families of logarithmic Sobolev inequalities, Poincar\'e inequalities and a gradient estimate of the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton Harnack inequality for the heat semigroup of the time dependent Witten Laplacian on manifolds equipped with a $(K, \infty)$-super Perelman Ricci flow. Based on a new second order time derivative formula on the Boltzmann-Shannon entropy for the heat equation of the Witten Laplacian, we introduce the $W_K$-entropy and prove its monotonicity for the heat equation of the Witten Laplacian on complete Riemannian manifolds satisfying $CD(K, \infty)$-condition and on compact manifolds equipped with a $(K, \infty)$-super Perelman Ricci flow. Our results characterize the $(K, \infty)$-Ricci solitons and the $(K, \infty)$-Perelman Ricci flows. We also use the $W_{K, m}$-entropy for the heat equation of the Witten Laplacian to characterize the $(K, m)$-Ricci solitons and the $(K, m)$-Perelman Ricci flows. Finally, we give a probabilistic interpretation of the $W_{K, m}$-entropy for the heat equation of the Witten Laplacian.