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$W$-entropy, super Perelman Ricci flows and $(K, m)$-Ricci solitons
- 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
- Subjects :
- Mathematics - Differential Geometry
Pure mathematics
Semigroup
010102 general mathematics
Monotonic function
01 natural sciences
symbols.namesake
Differential geometry
Differential Geometry (math.DG)
Fourier analysis
0103 physical sciences
symbols
FOS: Mathematics
Entropy (information theory)
Heat equation
010307 mathematical physics
Geometry and Topology
Mathematics::Differential Geometry
0101 mathematics
Laplace operator
Harnack's inequality
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....1664e5846de2baaa9963cf01e8480304
- Full Text :
- https://doi.org/10.48550/arxiv.1706.07040