Differential Harnack estimates for backward heat equations with potentials under geometric flows

Shouwen Fang; Peng Zhu

doi:10.3934/cpaa.2015.14.793

Article Contents

2015, Volume 14, Issue 3: 793-809. Doi: 10.3934/cpaa.2015.14.793

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Differential Harnack estimates for backward heat equations with potentials under geometric flows

Shouwen Fang^1, and
Peng Zhu^2,

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002
2.
School of Mathematics and physics, Jiangsu University of Technology, Changzhou, Jiangsu 213001

Received: December 31, 2013

Revised: November 30, 2014

Published: April 2015

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Abstract

In the paper we consider a closed Riemannian manifold $M$ with a time-dependent Riemannian metric $g_{i j}(t)$ evolving by $\partial_{t}g_{i j}=-2S_{i j}$ where $S_{i j}$ is a symmetric two-tensor on $(M,g(t))$. We prove some differential Harnack inequalities for positive solutions of backward heat equations with potentials when the metric satisfies the geometric flow. Some applications of these inequalities will be obtained. In particular, we show that the shrinking breathers of the Lorentzian mean curvature flow are the shrinking gradient solitons when the ambient Lorentzian manifold has nonnegative sectional curvature.

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Mathematics Subject Classification: Primary: 53C21, 53C44; Secondary: 35K05.

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