Let M be a closed Riemannian manifold with a Riemannian metric gij(t) evolving by a geometric flow ∂tgij = -2Sij, where Sij(t) is a symmetric two-tensor on (M, g(t)). Suppose that Sij satisfies the tensor inequality 2H(S, X)+E(S,X) ≥ 0 for all vector fields X on M, where H(S,X) and 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 Sij = Rij, 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 Müller (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.
- Differential Harnack Estimates
- Geometric flows
- Heat equations with potentials
- Ricci flow
ASJC Scopus subject areas
- Political Science and International Relations
- Geometry and Topology