Paper · Jun 19, 2017
-entropy, super Perelman Ricci flows and -Ricci solitons
As on arXiv
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.
- Submitted
- Jun 19, 2017
- Last revised
- Feb 25, 2018 (v2)
- Subjects
- math.DG
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
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.5Diffusions on Manifolds and Curvature
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12A.6𝓛-Geodesics and the Variational Formulas
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.