On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

Li, Songzi; Li, Xiangdong

doi:10.4310/ajm.2018.v22.n3.a10

Cited by 27 publications

(41 citation statements)

References 27 publications

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“…[26,27]) Under the same notation as in Theorem 3.1, defineH m,K (u, t) = − M u log udµ − m 2 (1 + log(4πt)) − m Kt…”

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W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

2018

Sci. China Math.

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In this survey paper, we give an overview of our recent works on the study of the W -entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W -entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W -entropy formula for the heat equation associated with the time dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the Wentropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result recaptures an important result due to Lott and Villani on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W -entropy formulas, we introduce the Langevin deformation of geometric flows on the cotangent bundle over the Wasserstein space and prove an extension of the W -entropy formula for the Langevin deformation. Finally, we make a discussion on the W -entropy for the Ricci flow from the point of view of statistical mechanics and probability theory.MSC2010 Classification: primary 58J35, 58J65; secondary 60J60, 60H30.It is natural and interesting to ask the problems what is the hidden idea for Perelman to introduce the mysterious W -entropy, what is the reason for him to call the quantity in (1) the W -entropy, and whether there is some essential link between the W -entropy and the Boltzmann entropy in statistical mechanics and probability theory.Inspired by Perelman [44] and related works [39,40], the second author of this paper proved in [20] the W -entropy formula for the heat equation of the Witten Laplacian on compact Riemannian manifolds with the CD(0, m)-condition and gave a probabilistic interpretation of the W -entropy for the Ricci flow. Later, the W -entropy formula and a rigidity theorem for the W -entropy were proved in [22,24] for the fundamental solution to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)condition, and the W -entropy formula was proved in [21] for the Fokker-Planck equation of the Witten Laplacian on complete Riemannian manifolds with the CD(0, m)-condition. The relationship between Perelman's W -entropy formula for the Ricci flow and the Boltzmann H-theorem for the Boltzmann equation was discussed in [23] from the point of view of the statistical mechanics. In [26,27,29,30,31], we extended the W -entropy formula to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)condition and on compact Riemannian manifolds equipped with (K, m)-super Ricci flows. Moreover, we proved in [28,31] an analogue of the W -entropy formula for the geodesic flow on the Was...

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“…[26,27]) Under the same notation as in Theorem 3.1, defineH m,K (u, t) = − M u log udµ − m 2 (1 + log(4πt)) − m Kt…”

mentioning

confidence: 99%

“…[27,30,31]) Let (M, g(t), φ(t), t ∈ [0, T ]) be a compact (K, ∞)-super Ricci flow satisfying the conjugate heat equation(16).Let u(·, t) = P t f be a positive solution to the heat equation ∂ t u = Lu with u(·, 0) = f , where f is a positive and measurable function on M . Define…”

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confidence: 99%

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

2018

Sci. China Math.

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“…This paper is an improved version of a part of our previous preprint [14]. Due to the limit of the length of the paper, we split [14] into several papers. See also [15,16,17].…”

Section: Theorem 24mentioning

confidence: 99%

“…Following [1], we say that (M, g, µ) satisfies the curvature-dimension CD(K, m)-condition 1 for a constant K ∈ R and m ∈ [n, ∞] if and only if See also our previous paper [13] and [14,15,16]. When φ is a constant and m = n, the (K, n)- (…”

Section: Introductionmentioning

confidence: 99%

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

2018

Journal of Functional Analysis

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“…To end this section, let us mention that this paper is a revised version of a part of our 2014-2016 preprint [15], which contained also the Li-Yau and the Li-Yau-Hamilton type Harnack inequalities on variants of (K, m)-super Perelman Ricci flows. As the preprint [15] is too long, we have divided it into three papers (see also [16,17]).…”

Section: Introductionmentioning

confidence: 99%

W-Entropy, Super Perelman Ricci Flows, and (K, m)-Ricci Solitons

2019

J Geom Anal

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In this paper, we prove the characterization of the (K, ∞)-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, ∞)-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, ∞)-condition and on compact manifolds with (K, ∞)-super Perelman Ricci flows. Our results characterize the (K, ∞)-Ricci solitons and the (K, ∞)-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. MSC2010 Classification: primary 53C44, 58J35, 58J65; secondary 60J60, 60H30. L = ∆ − ∇φ · ∇ is a self-adjoint and non-positive operator on L 2 (M, µ). For all u, v ∈ C ∞ 0 (M ), the following integration by parts formula holds M ∇u, ∇v dµ = − M Luvdµ = − M uLvdµ.

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On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

Cited by 27 publications

References 27 publications

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

W-Entropy, Super Perelman Ricci Flows, and (K, m)-Ricci Solitons

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