Harnack Estimates for Nonlinear Backward Heat Equations in Geometric Flows

Abstract

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow ${\partial }_t{g}_{ij}=-2S_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation \begin{eqnarray*} \frac{\partial f}{\partial t} = -Δf + γf\log f +aSf \end{eqnarray*} where $a$ and $\gamma$ are constants and $S=g^{ij}{S}_{ij}$ is the trace of $S_{ij}$. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover lead to new Harnack inequalities for a variety of geometric flows.

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