等距嵌入及其在广义相对论中的应用

等距嵌入是一个具有悠久历史的课题，在微分几何和偏微分方程领域都有基础地位的重要性，在数学广义相对论中也有广泛的应用。本项目将研究几个典型的等距嵌入问题和广义相对论中刻画拟局部质量的几何不等式及其刚性：.1 Schwarzschild空间中的Weyl问题;.2 静态流形中的Riemann Penrose不等式;.3 Alexandrov-Nirenberg曲面在R^3中的等距嵌入。.申请人和Miami大学苗蓬子副教授、复旦大学王志张教授等人合作，对以上问题有充分的前期准备和研究基础；在本项目的实施过程中，拟综合使用微分几何、积分几何以及微分方程中的各类技巧，包括几何变分方法、积分方法、极值原理、非线性迭代、紧性方法、嵌入定理等。

Isometric embedding is a long-standing topic which has assumed a position of fundamental importance in differential geometry and PDEs. Moreover it's applied extensively to mathematical relativity. This project is concerned with the following problems of isometric embedding and some geometric inequalities related to quasi local mass:. 1 The Weyl problem in Schwarzschild spaces. The applicant joints with Professor Zhizhang Wang in Fudan University to obtain the openness in method of continuity if the ambient space is three dimensional.. 2 Riemann Penrose inequality in static manifold. The applicant joints with Associate Professor Pengzi Miao in Miami University and Zhizhang Wang in Fudan University to establish the rigidity of isometric embedding in rotational symmetry static manifold, which helps study the case of equality of Riemann Penrose inequality and characterize the structure of solution to Einstein equation in static spacetime.. 3 The isometric embedding of Alexandrov-Nirenberg surfaces in R^3. The applicant discusses the geometric aspects of the surfaces, and elaborates the structure of solutions to the linearized equations.. In this project, classical techniques in differential geometry, integral geometry and geometric PDEs will be utilized and developed, such as geometric calculus of variations, integral method and maximum principle, nonlinear iteration, compactness method and embedding theorem.