Harnack estimates for the linear heat equation under the Ricci flow

Abstract

We consider the linear heat equation on a manifold that evolves under the Ricci flow. The gradient estimates for positive solutions as well as Li‐Yau type inequalities are given in this paper. Both the case where M is a complete manifold without boundary and the case where M is compact are considered. We have also obtained the Harnack inequalities for the heat equation on M by previous results. The heat equation is a classical subject that has been extensively studied and has lead to many important results, especially in studies of differential geometry. One of the important techniques used in studying the heat equation is the differential Harnack inequality developed by Li and Yau [1986]. This is also applied to Ricci flow by Hamilton [1993], and plays an important role in solving the Poincare conjecture. We consider the positive solutions of the linear heat equation on a manifold M that evolves under the Ricci flow. A series of gradient estimates are obtained for such solutions, including several Li‐Yau-type inequalities. The manifold M considered here is a complete manifold without boundary. Let M be a manifold without boundary, and .M; g.x; t//t2T0;TU be a complete solution to the Ricci flow