Vol. 278, No. 1, 2015

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

Songzi Li and Xiang-Dong Li

Vol. 278 (2015), No. 1, 173–199
Abstract

We develop a new approach to prove the W-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds, giving a natural geometric interpretation of the quantities appearing in the W-entropy formula. We also prove the W-entropy formula for the Witten Laplacian on compact Riemannian manifolds with time dependent metrics and potentials, as well as for the backward heat equation associated with the Witten Laplacian on compact Riemannian manifolds equipped with Lott’s modified Ricci flow. Our results extend to complete Riemannian manifolds with negative m-dimensional Bakry–Émery Ricci curvature, and to compact Riemannian manifolds with K-super m-dimensional Bakry–Émery Ricci flow. As an application, we prove that the optimal logarithmic Sobolev constant on compact manifolds equipped with the K-super m-dimensional Bakry–Émery Ricci flow is decreasing in time.

Keywords
Bakry–Émery Ricci curvature, $W$-entropy, Witten Laplacian, modified Ricci flow
Mathematical Subject Classification 2010
Primary: 53C44, 58J35, 58J65
Secondary: 60J60, 60H30
Milestones
Received: 2 November 2014
Revised: 24 March 2015
Accepted: 25 March 2015
Published: 30 September 2015
Authors
Songzi Li
School of Mathematical Sciences
Fudan University
220 Handan Road
Shanghai 200433
China
Institut de Mathématiques de Toulouse
Université Paul Sabatier
118 route de Narbonne
31062 Toulouse Cedex 9
France
Xiang-Dong Li
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
55 Zhongguancun East Road
Beijing 100190
China