$W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials

Abstract

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, and derive the $W$-entropy formula for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott's modified Ricci flow. We also extend our results to complete Riemannian manifolds with negative $m$-dimensional Bakry-Emery Ricci curvature, and to compact Riemannian manifolds with $K$-super $m$-dimensional Bakry-Emery Ricci flow. As application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the $K$-super $m$-dimensional Bakry-Emery Ricci flow is decreasing in time.

This is an extended version of the previous version. Section $6$ and Remark $2.2$ are added

References in corpus (3)

• $W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials
• Harnack inequalities and $W$-entropy formula for Witten Laplacian on Riemannian manifolds with $K$-super Perelman Ricci flow
• Hamilton's Harnack inequality and the $W$-entropy formula on complete Riemannian manifolds

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• On the Shannon entropy power on Riemannian manifolds and Ricci flow
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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
• Heat kernel estimates and the essential spectrum on weighted manifolds
• 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
• Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry
• Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry
• Laplacian comparison theorem on Riemannian manifolds with modified m-Bakry-Emery Ricci lower bounds for $m\le 1$
• $W$-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds