Harnack Estimates for Nonlinear Backward Heat Equations in Geometric Flows

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, 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} = -{\Delta}f + \gamma f\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

Harnack Estimates for Nonlinear Heat Equations with Potentials in Geometric Flows

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential: \begin{eqnarray*} \frac{\partial f}{\partial t} = {\Delta}f + \gamma (t) f\log f +aSf, \end{eqnarray*} where $\gamma (t)$ is a continuous function on $t$, $a$ is a constant and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows

Perelman's Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds

In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + infinity whenever the Yamabe invariant is positive.Comment: LaTeX2e, 7 pages. To appear in Arch. Math. Revised version improves result to also cover positive cas