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

Abstract

In this paper, we prove the characterization of the $(K,\infty )$-super Perelman Ricci flows by various 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 second order differential inequality on the Boltzmann-Shannon entropy 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 a second order differential entropy inequality on $(K,m)$-super Ricci flows, which can be used to characterize the $(K,m)$-Ricci solitons and the $(K,m)$-Ricci flows. Finally, we give a probabilistic interpretation of the $W$-entropy for the heat equation of the Witten Laplacian on manifolds with the $CD(K,m)$-condition.

We remove Section 5 from the previous version and add two new results in Section 5. arXiv admin note: text overlap with arXiv:1412.7034

References in corpus (2)

• Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow
• Weak solutions for the Ricci flow I