Fully nonlinear equations in geometry and general relativity

点击翻译按钮获取中文摘要

The mathematics I will explore in this research program are taken from various problems arising from geometry and general relativity. Since the universe of geometry and general relativity is highly nonlinear, I will study these problems from the point of fully nonlinear partial differential equations. I will mainly focus on the following three short-term objectives in the program. My first short-term objective is to study isometric embedding and quasi-local mass. This is the continuation of my previous work. One of the most important problems I am concerning is to establish isometric embedding into Schwarzschild manifold homologous to the horizon. The successful progress of this problem will lead to the localized Riemannian Penrose inequality, which is closely related to cosmic censorship conjecture in general relativity. My second short-term objective is to study the geometry of asymptotically hyperbolic manifolds. These manifolds come naturally from the study of general relativity with negative cosmological constant. It is also closely related to the study of null geometry in general relativity. One of the most important questions in this area is whether anti-de Sitter-Schwarzschild manifold is the ”best” model manifold in this class. To answer this question, I will first establish the rigidity of anti-de Sitter-Schwarzschild manifold among the class of static metrics. I will then study the capacity of asymptotically hyperbolic manifolds. The long time goal is to establish the Riemannian Penrose inequality in this setting. My third short-term objective concerns the fully nonlinear equations invariant under Mobius transformation in dimension 2. It is the natural extension of Yamabe problem in dimension 2. I will study the existence and uniqueness for such equations and their applications in geometry, particularly in conformal geometry. The successful progress in my projects will help us to understand the universe of geometry and general relativity. It will also shed lights on other areas of mathematics.