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

Songzi Li,Xiang-Dong Li

doi:10.2140/pjm.2015.278.173

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.

Highlights

Let M be a complete Riemannian manifold with a fixed Riemannian metric g and a fixed potential φ ∈ C2(M)
The purpose of this paper is to extend the W-entropy formula in Theorem 1.1 to the heat equation (1) associated with the time dependent Witten Laplacian on compact Riemannian manifolds equipped with time dependent metrics and potentials
As an application of the W-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials, in the following theorem we prove that the optimal logarithmic Sobolev constant associated with the Witten Laplacian on compact manifolds equipped with the m-dimensional Perelman super Ricci flow is decreasing in time

Summary

Introduction

Let M be a complete Riemannian manifold with a fixed Riemannian metric g and a fixed potential φ ∈ C2(M). In the case where (M, g) is a complete Riemannian manifold with the bounded geometry condition, to [Lott 2003; Charalambous and Lu 2015], by introducing a sequence of warped product metrics {gε} on M = M × N defined by gε = g ⊕ ε2e−2φ/(m−n)gN , and using the fact that the heat kernel of the Laplace–Beltrami (M,gε) on (M, gε) (with renormalized volume measure) converges in the C2,α ∩ W 2,p-topology to the heat kernel of the Witten Laplacian L = M − ∇φ · ∇ on (M, g, μ), we can use the same approach as in the compact case to give a new proof of the W-entropy formula for the heat kernel of the Witten Laplacian on complete Riemannian manifolds satisfying the bounded geometry condition in Theorem 1.1 We will study this problem in detail in the future. See related works of Y.-Z. Wang et al [2013; 2014]

Dissipation formula of the Boltzmann–Shannon entropy

The W-entropy for the Ricci flow on warped product manifolds