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

Authors :
Songzi Li
Xiang-Dong Li
Publication Year :
2017
Publisher :
arXiv, 2017.

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.<br />Comment: 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

Details

Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....1664e5846de2baaa9963cf01e8480304
Full Text :
https://doi.org/10.48550/arxiv.1706.07040