Geometric flows and differential Harnack estimates for heat equations with potentials

Abstract

Let \(M\) be a closed Riemannian manifold with a Riemannian metric \(g_{ij}(t)\) evolving by a geometric flow \(\partial _{t}g_{ij} = -2{S}_{ij}\), where \(S_{ij}(t)\) is a symmetric two-tensor on \((M, g(t))\). Suppose that \(S_{ij}\) satisfies the tensor inequality \(2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0\) for all vector fields \(X\) on \(M\), where \({\mathcal H}(S, X)\) and \({\mathcal E}(S,X)\) are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where \(S_{ij} = R_{ij}\), the Ricci tensor of \(M\), our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Muller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.