Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow

Abstract

In this paper, we prove logarithmic Sobolev inequalities and derive the Hamilton Harnack inequality for the heat semigroup of the Witten Laplacian on complete Riemannian manifolds equipped with $K$-super Perelman Ricci flow. We establish the $W$-entropy formula for the heat equation of the Witten Laplacian and prove a rigidity theorem on complete Riemannian manifolds satisfying the $CD(K,m)$ condition, and extend the $W$-entropy formula to time dependent Witten Laplacian on compact Riemannian manifolds with $(K,m)$-super Perelman Ricci flow, where $K\in \mathbb{R}$ and $m\in [n,\infty ]$ are two constants. Finally, we prove the Li-Yau and the Li-Yau-Hamilton Harnack inequalities for positive solutions to the heat equation ${\partial }_{t}u=Lu$ associated to the time dependent Witten Laplacian on compact or complete manifolds equipped with variants of the $(K,m)$-super Ricci flow.

Theorem 1.1 in the first version has been improved. In the case of time dependent metrics and potentials, an error in the proof of Theorem 1.5 (i.e., Theorem 2.2), Theorem 2.3 and Theorem 2.4 in the first version has been corrected. See Section 4 and Section 5

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• Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces
• $W$-entropy, super Perelman Ricci flows and $(K,m)$-Ricci solitons
• Hamilton differential Harnack inequality and $W$-entropy for Witten Laplacian on Riemannian manifolds
• On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows
• $W$-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds