Researches On Matrix Li-Yau-Hamilton Estimates

This paper consists of two main research objects,one is the differential Harnack inequality,and the other is the frequency functional monotonicity.In Chapter 2 I briefly reviewed the classical methods of functional gradient estimates belongs to Li-Yau and introduced a new method to verify the gradient estimation of Li-Yau.Inspired by this thinking,combined with Hamilton’s tensor-type maximum principle,I discussed a class of matrix Li-Yau-Hamilton estimates for partial differential equation solutions on Riemannian manifolds(Chapter 3),K¨ahler manifolds,and K¨ahler-Ricci flows(Chapter 4),and got some good results.When we take some specific equations in the class,we can get some inferences,and the resulting inferences are consistent with the results obtained by the predecessors.In particular,for a particular equation on a Riemannian manifold,we can obtain matrix-type estimates with a positive lower bound through more elaborate techniques.The monotonicity of the frequency functional can be seen as an application of the differential Harnack inequality.In the 5th chapter,we construct three types of frequency functionals on Riemannian manifolds,surfaceric Ricci flow,and K¨ahler-Ricci flow respectively.Later,with the help of some existing matrix-type differential Harnack inequalities,the monotonicity of these three types of frequency functionals with respect to time was proved.In particular,we summarize several different frequency function for Riemannian manifolds and Ricci flow on surface with an equation,simplifying the proof process of predecessors to some extent,while also generalizing the results of predecessors.