$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons
Abstract
In this paper, we prove the characterization of the -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 -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 -entropy quantity and prove its monotonicity for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the -condition and on compact manifolds with -super Perelman Ricci flows. Our results characterize the -Ricci solitons and the -Perelman Ricci flows. We also prove a second order differential entropy inequality on -super Ricci flows, which can be used to characterize the -Ricci solitons and the -Ricci flows. Finally, we give a probabilistic interpretation of the -entropy for the heat equation of the Witten Laplacian on manifolds with the -condition.
Keywords
Cite
@article{arxiv.1706.07040,
title = {$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons},
author = {Songzi Li and Xiang-Dong Li},
journal= {arXiv preprint arXiv:1706.07040},
year = {2018}
}
Comments
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