Geometric Partial Differential Equations in General Relativity

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The goal of this proposal is to understand the Einstein constraint equations and to explore new geometric properties of asymptotically flat manifolds. The first part of the project continues the studies of the PI and co-PI on hypersurfaces with nonnegative scalar curvature. They propose to classify scalar-flat hypersurfaces and to apply the arguments to study the ADM mass inequalities. The second part of the project studies the density theorems to the full Einstein constraint equations and their relations to the global conserved quantities, including center of mass and angular momentum. The proposed research will also have applications to the complex Hessian equations which generalize the Calabi-Yau equations on Kaehler manifolds.The project studies several fundamental problems to better understand the analytical and geometric structure of the mathematical models of our universe. The proposed techniques connect different branches of mathematics and physics, including general relativity, Riemannian geometry, complex geometry, and partial differential equations. The new ideas have the potential for setting long-standing questions in classical differential geometry. Moreover, the proposal will generate research problems suitable for undergraduate and graduate students and lead to research activities in the New England area to attract more students into the current active research fields.