Geometric Problems in General Relativity

PI计划研究与微分几何和广义相对论相关的问题。第一个项目涉及渐近平面流形的基本问题，特别是质心和常平均曲率叶理。证明了哈密顿质心在时间片上的定义，构造了几何中心等于质心的常平均曲率叶理。PI计划进一步研究质心在时空中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对于理解渐近平面流形的内在几何是重要的。即使在哈密顿质心可能无法定义的情况下，PI也将继续研究叶状的存在。此外，PI先前提出的一个新的密度定理表明，调和渐近解在爱因斯坦约束方程解的空间中是泛型的。她打算用这个定理进一步研究解的物理量，如角动量。第二个项目涉及在各种环境空间中具有边界的紧致流形。吴大民的PI在欧几里德空间和双曲空间中发展了具有边界的超曲面在半球上的刚性结果。她计划在几种不同曲率条件下，对球面上具有边界的超曲面和欧几里德空间中具有更高余维的子流形提出类似的刚性定理。PI的项目将有助于更好地理解物理量及其与渐近平面流形的几何关系，渐近平面流形是广义相对论中孤立系统的模型。特别地，常平均曲率叶理为孤立系统提供了一个固有坐标系。在二元黑洞问题中，叶理将有助于理解黑洞的相互作用，这是理论和数值物理学的基础。她还认为，规范坐标系有助于不同群体的数字学家比较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其余工作自然出现在广义相对论和各种物理情况中。对平面区域和球面等模型情况进行表征和分类是十分重要的。这些几何对象的刚度结果对分类是必不可少的。

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.