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

Abstract

Abstract. In this paper, we prove the Hamilton differential MSC2010 Classification: primary 53C44, 58J35, 58J65; secondary 60J60, 60H30.

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

“…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 Wasserstein space over Riemannian manifolds with the CD(0, m)-condition, which recaptures an important result due to Lott and Villani [34,33] on the displacement convexity of the Boltzmann-Shannon entropy on the Wasserstein space over Riemannian manifolds with non-negative Ricci curvature.…”

“…[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…”

The concavity of p-Rényi entropy power for doubly nonlinear diffusion equations and L-Gagliardo-Nirenberg-Sobolev inequalities

“…In [9], Li-Li prove an entropy power inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with CD(K, m)-condition and on compact manifolds equipped with (K, m)-super Ricci flows. In this paper, we consider the connection between Rényi entropy and the doubly nonlinear diffusion equation(DNDE)…”

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

“…In time independent case, we pointed out in [17] a close and deep connection between the Li-Yau-Hamilton type Harnack inequality and the W -entropy for the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)-condition.…”

“…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]).…”