Geometric flows and differential Harnack estimates for heat equations with potentials

Let M be a closed Riemannian manifold with a Riemannian metric g i j ( t ) evolving by a geometric flow ∂ t g i j = - 2 S i j , where S i j ( t ) is a symmetric two-tensor on ( M , g ( t ) ) . Suppose that S i j satisfies the tensor inequality 2 H ( 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 S i j = R i j , 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.