Harnack Estimates For Geometric Flows, Applications To Ricci Flow Coupled With Harmonic Map Flow

The Harnack inequality of geometric flow, also known as Li- Yau-Hamilton inequality, Plays a very important role in geometric analysis.First of all,Parabolic equation Harnack inequality stemmed from Moser’s work in 1964, He studied the linear type divergence equation,And made breakthrough progress. Li and Q use maximum principle to get flow on the heat equation Harnack inequality in 1986, This is the first time the differential equation of combined Harnack inequality and geometry.Hamilton, then use the same technique(The maximum principle)has been considered on some Harnack inequality of nonlinear equations. In recent years, The research about the Harnack inequality geometric flow is very much also.In this article, first of all, we get hot in the abstract geometrical flow equation and the equation of conjugate heat Harnack estimates, Then according to the results we can get new Harnack inequality of different geometric flow, especially the Ricci flow manifold combination of harmonic mapping Harnack inequality.Of course we will need to carry on the simple introduction about the important knowledge, and need the necessary research, draw the conclusion that can be used for paper. Finally, the core of the thesis is derived.The paper specific arrangement is as follows:The first chapter: Overview of the geometric flow Harnack inequality and the current research status.The second chapter: The basic principle of Ricci manifold equation The third chapter: The maximum principle on the heat equation The fourth chapter: The heat equation on the Li- Yau estimates.The fifth chapter: On the surface of the Harnack estimates with c >0.The sixth chapter: Linear insert Harnack estimation.The seventh chapter: The proof of the matrix Harnack estimates.The eighth chapter: Geometric flow Harnack estimates and the Ricci manifold coupled with the application of harmonic map flow.