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

Abstract

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.

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.…”

“…In our recent paper [29], we gave a probabilistic interpretation of the W m,K -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(K, m)-condition. More precisely, let m ∈ N, M = R m , g 0 the Euclidean metric,…”

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

“…Due to the limit of the length of the paper, we split [14] into several papers. See also [15,16,17].…”

“…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)- (…”

“…In our previous papers [13,16], we proved the W -entropy formula for the heat equation of the Witten Laplacian on complete Riemannian manifolds with the CD(−K, m)-condition, m ∈ [n, ∞) and K ≥ 0. These extend Ni and Li-Xu's results from the standard case of heat equation of the Laplace-Beltrami operator on complete Riemannian manifolds with Ricci curvature condition to the general case of the heat equation of the Witten Laplacian on complete weighted Riemannian manifolds with suitable m-dimensional Bakry-Emery Ricci curvature condition.…”