Harnack estimates for conjugate heat kernel on evolving manifolds

In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation provides a unified framework for some known results, in particular including corresponding results of Ni (J Geom Anal 14(1): 87-100, 2004), Perelman (arXiv: math. DG/0211159, 2002) and Tran (arXiv: 1211.6448, 2012) as special cases. Moreover, it leads to new results in the setting of Ricci-Harmonic flow and mean curvature flow in Lorentzian manifolds with nonnegative sectional curvature.