数学广义相对论与共形几何中的若干问题研究

One of the most important application of differential geometry in history lies in General Relativity. For example, one major object in mathematical General Relativity is asymptotically flat manifold, by means of which one can describe an isolated gravitational system. We can define mass and centre of mass for the asymptotically flat end. On the other hand, it has plenty of geometric properties. For example, one can study constant mean curvature surfaces and isoperimetric surfaces in asymptotically flat manifolds. And similar problems also arise in asymptotically hyperbolic manifolds. Asymptotically flat manifolds and asymptotically hyperbolic manifolds have close relationship to conformal geometry.. The aim of this project is to study asymptotically flat manifolds and asymptotically hyperbolic manifolds and extensions will be made to.conformal geometry. The following problems are included. Properties of constant mean curvature surfaces and isoperimetric surfaces in asymptotically flat and asymptotically hyperbolic manifolds including the existence, uniqueness and stability of such surfaces; Locally conformally flat manifolds, including the injectivity of developing map, and the dimension of the boundary of developing domain; Hypersurfaces in hyperbolic space form, including when immersion implies embedding and the classification of embedded surfaces under certain convex conditions; Description of locally conformally flat manifolds with nonnegative Ricci curvature, etc.

历史上微分几何最重要的应用之一就是在广义相对论中。比如渐近平坦流形就是广义相对论中一个最基本的研究对象，我们可以用它来描述一个孤立的引力系统，我们可以对一个渐近平坦的末端定义质量和质心的概念。另外，它又有丰富的几何性质，比如研究其中的常平均曲率曲面和等周曲面。同样，也可以在渐近双曲流形中考虑类似的问题。另外，渐近平坦（双曲）流形又跟共形几何有着密切的关联。. 本项目旨在对渐近平坦（双曲）流形几何性质做系统的研究，并且延伸到共形几何领域。主要问题包括，渐近平坦流形和渐近双曲流形中的常平均曲率曲面和等周曲面的性质，包括这些曲面的存在性，唯一性，稳定性等；局部共形平坦流形的性质，主要包括展开映射的单射性，和展开映射像集边界维数问题；双曲空间形式中的超曲面性质，主要包括在一些凸性的假设下，浸入超曲面是否是嵌入，以及嵌入超曲面的分类问题；Ricci曲率非负的局部共形平坦流形的刻画等。

本项目主要研究数学广义相对论中与共形几何中的若干问题。在本项目的研究过程中，项目负责人和参与人主要研究了：1. 双曲空间中的凸超曲面；2. 完备非紧Ricci曲率非负的局部共形平坦流形；3. Huber型定理的高维推广等一系列问题。项目负责人和参与人在本项目资助下，在国内外知名数学期刊上共发表7篇文章，并且有3篇预印本待发表。