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

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.