Harnack estimates for nonlinear backward heat equations in geometric flows

Let M be a closed Riemannian manifold with a family of Riemannian metrics gij(t) evolving by a geometric flow ∂tgij=−2Sij, where Sij(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∂f∂t=−Δf+γflog⁡f+aSf where a and γ are constants and S=gijSij is the trace of Sij. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover leads to new Harnack inequalities for a variety of geometric flows.