几何分析是运用分析的方法来研究几何问题，是当前数学中很活跃的一个领域。几何流将流形的曲率和微分方程很好地结合起来，成为几何分析中很重要的一种工具，是近二十年来几何分析中最活跃的方向之一。本项目主要在紧致流形和Warped乘积流形上研究延拓的Ricci流和一般几何流下一些热方程和反向热方程正解的一类微分Harnack不等式及其应用问题。我们拟使用抛物方程的最大值原理、黎曼几何中的Bochner公式及Warped乘积流形理论，期望得到这些热方程的正解的微分Harnack不等式以及一些梯度估计，给出对应的熵公式，利用熵的单调性得到这些几何流的奇点的几何性质。. 通过对这些问题的研究，可以更好地描述在一定曲率条件下流形的几何结构。

Geometric analysis is to research into geometry by using the methods of analysis and has become a very active field in mathematics at present. The geometric flows which combine well differential equations with the curvature in manifolds have become an important tool in geometric analysis and one of the most active fields in geometric analysis in recent twenty years. In this project we mainly study a class of differential Harnack inequalities for the positive solutions of some heat equations and backward heat equations under the extended Ricci flow and the generic geometric flow and their applications on compact manifolds and warped product manifolds. We will use the maximum principle of parabolic equations, Bochner formula in Riemannian geometry and the theory of warped product manifolds. Some differential Harnack inequalities and gradient estimates for positive solutions to these heat equations will be obtained. The associated entropy formulas will be given and the geometric properties for the singularity of these geometric flows will also be followed from the monotonicity of these entropies.. The geometric structures on manifolds with some curvature conditions will be described better by studying them.