广义相对论中关于共形紧流形的若干几何问题

Conformally compact manifolds are a kind of important research objects in view of AdS/CFT correspondence in general relativity. Many physical models, including Anti-deSitter spacetime, Schwarzschild-AdS black hole and AdS soliton, are all conformally compact manifolds. In this project, we will mainly concentrate on the geometrical relation between conformally compact manifold and its conformal boundary-at-infinity, and consider the following problems:.(1)the asymptotic expansion near the conformal infinities as well as the rigidity of 4-dimensional conformally compact Bach flat manifolds with constant scalar curvature;.(2)the proposal of a new kind of mass on asymptotically conformally compact Einstein manifolds and the behavior of this mass under Ricci flow;.(3)the property of the Horowitz-Myers geon.. The results of this project will help to understand the nature of the above physical models and stimulate the development of research on related problems.

共形紧流形是基于广义相对论中AdS/CFT对应所提出的一类重要研究对象，许多物理模型，包括反德希特时空，史瓦西-反德希特黑洞，反德希特孤子等均为共形紧流形。本项目拟围绕共形紧流形与其共形无穷之间的几何关系，研究如下问题：.（1）具有常数量曲率的4维共形紧巴赫平坦流形在共形无穷附近的展开以及刚性；.（2）渐近共形紧爱因斯坦流形上一类新质量的提出以及新质量在里奇流下的行为；.（3）Horowitz-Myers真子的性质。. 本项目的研究成果有助于了解上述物理模型的本质，促进广义相对论中相关问题的发展。

本项目围绕广义相对论中共形紧流形与其共形无穷之间的几何关系，主要开展两个课题的研究，课题一是4维共形紧Bach平坦流形刚性的研究，课题二是渐近共形紧Einstein流形质量的研究。利用共形无穷附近的渐近展开以及对方程精细的估计，我们得到了具有调和曲率的4维渐近Poincaré-Einstein流形的刚性，加深了对4维共形紧Bach平坦流形的渐近双曲质量的了解。我们还梳理了Bartnik数据的几何与拟局域质量、正质量定理、非负数量曲率填充之间的关系，为广义相对论中质量和正质量定理的研究增加了很多新的问题。