Some Geometric Analysis On Smooth Metric Measure Spaces And Harmonic Ricci Flow

This thesis mainly concentrates on some qualitative geometric analysis on smooth metric measure spaces and harmonic Ricci flow, including gradient estimates and en-tropy formulae for some nonlinear diffusion equations and harmonic equation, eigen-value estimates, differential Harnack inequalities and variational formulae. The basic tools used here consist maximum principle, Moser iteration and Bochner techniques etc. More details are as follows,In chapter1we give a survey to the motivation and recent research progress on geometric evolution equations, as well as some questions what we will discuss in this thesis. The second chapter is the preliminaries, involved the necessary concepts and formulae, such as Bakry-Emery Ricci curvature, weighted Bochner formula and basic variational formula related to harmonic Ricci flow.The third chapter is the main part of thesis which includes three nonlinear diffu-sion equations:porous medium and fast diffusion equation, p-Laplacian heat equation and doubly degenerate diffusion equation. Firstly, we obtain global and local Aronson-Benilan type estimate for weighted porous medium equation, a Hamilton type elliptic estimate for weighted fast diffusion equation with m-Bakry-Emery Ricci curvature bounded below on smooth metric measure spaces. Secondly, an optimal global Li-Yau type gradient estimate and entropy formula for weighted p-Laplacian heat equa-tion are derived under the condition of nonnegative m-Bakry-Emery Ricci curvature, moveover, we get a local gradient estimate for weighted p-Laplacian equation by means of Moser iteration technique and the first eigenvalue estimate for weighted p-Laplacian operator. Thirdly, an optimal gradient estimate and entropy monotonicity formula are established for the solutions of doubly degenerate diffusion equation with nonneg-ative Ricci curvature on general Riemannian manifold, the analogue results are valid for weighted case.In chapter4we discuss a variety of monotonicity formulae in geometric analysis, also show some generalization of three new monotonicity formulae in recent paper of Colding about Green function to f-harmonic function.The fifth chapter includes some reviews about the differential Harnack inequal-ities of geometric evolution equation, then shows a Li-Yau-Hamilton inequality and a constrained interpolating Harnack estimate for solutions of a semilinear parabolic equation. Furthermore, we investigate the second variation formula for gradient har-monic Ricci soliton and the first variational formula for weighted Gibbons-Hawking-York functional under the mean curvature flow in the harmonic Ricci flow background.