Geometric Problems in General Relativity

广义相对论中的几何问题

• 批准号: 1005560
• 负责人: Lan-Hsuan Huang
• 金额: $ 12.56万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2012-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005560&HistoricalAwards=false
• 关键词: Geometric; Problems; General; Relativity

The PI is proposing to study problems which relate to both differential geometry and general relativity. The first project concerns the fundamental problems in asymptotically flat manifolds, particularly center of mass and constant mean curvature foliations. The PI has proved that the notion of Hamiltonian center of mass is well-defined in the time-slice and has constructed the constant mean curvature foliation whose geometric center is equal to the center of mass. The PI is proposing to further investigate the properties of center of mass in spacetime and the time evolution of center of mass and the foliation. The existence of constant mean curvature foliation is important for understanding the intrinsic geometry of asymptotically flat manifolds. The PI will continue to study the existence of the foliation even when the Hamiltonian center of mass might not be defined. In addition, a new density theorem previously developed by the PI shows that the solutions with harmonic asymptotics are generic in the space of solutions to the Einstein constraint equations. She intends to use the theorem to further study the physical quantities of the solutions, such as angular momentum. The second project deals with compact manifolds with boundary in various ambient spaces. The PI with Damin Wu developed the rigidity results on hemispheres for hypersurfaces with boundary in either Euclidean space or hyperbolic space. She plans to develop the analogous rigidity theorems for hypersurfaces with boundary in the sphere and for submanifolds in Euclidean space with higher codimensions under several different curvature conditions.The PI's projects will lead to a better understanding of the physical quantities and their connection to geometry in asymptotically flat manifolds, which model the isolated systems in general relativity. In particular, the constant mean curvature foliation provides an intrinsic coordinate system for isolated systems. The foliation will be useful to understand the interaction of black holes in the problem of binary black holes which is fundamental in both theoretical and numerical physics. She also believes that the canonical coordinate system is helpful for numerists from different groups to compare their results established under different coordinate systems. The remainder of the PI's proposed work about compact regions with boundary naturally arises in general relativity and in various physical situations. It is fundamental to characterize and to classify the model cases, such as the flat regions and spheres. The rigidity results on these geometric objects are essential for the classification.

PI建议研究与微分几何和广义相对论有关的问题。第一个项目涉及渐近平坦流形的基本问题，特别是质心和常平均曲率叶理。 PI证明了哈密顿质心的概念在时间片上是明确的，并构造了几何中心等于质心的常平均曲率叶理。PI建议进一步研究时空中质心的性质以及质心和叶理的时间演化。常平均曲率叶理的存在性对于理解渐近平坦流形的内在几何是很重要的。PI将继续研究叶理的存在，即使在哈密顿质心可能没有定义的情况下。此外，PI先前开发的一个新的密度定理表明，具有调和渐近性的解在爱因斯坦约束方程的解空间中是通用的。她打算利用该定理进一步研究解的物理量，如角动量。第二个项目涉及各种环境空间中的边界紧流形。PI与吴大民发展了半球面上具有边界的超曲面在欧氏空间或双曲空间中的刚性结果。她计划在几种不同的曲率条件下，为具有球面边界的超曲面和具有更高余维的欧氏空间中的子流形建立类似的刚性定理。PI的项目将有助于更好地理解渐近平坦流形中的物理量及其与几何的联系，这些流形模拟了广义相对论中的孤立系统。特别地，常平均曲率叶理为孤立系统提供了一个内在的坐标系。叶理将有助于理解黑洞的相互作用的问题，这是在理论和数值物理的基础。她还认为，正则坐标系有助于不同群体的数字学家比较他们在不同坐标系下建立的结果。PI提出的关于有边界的紧致区域的工作的其余部分自然出现在广义相对论和各种物理情况中。它是基本的特征和分类的模型情况下，如平坦区域和球体。这些几何对象上的刚度结果对于分类是必不可少的。