Geometric Problems in General Relativity

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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.