Geometric Compactness Theorems with Applications to General Relativity

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In General Relativity, spacetime and spacelike slices of spacetime are manifolds (objects that locally resembles Euclidean spaces) satisfying certain geometric conditions determined by the Einstein Equation and other physically natural constraints. The manifolds arising in General Relativity are curved by gravity and they can contain black holes or thin deep gravity wells, making it technically difficult to estimate how close the manifold is to a simplified model, like Euclidean space. New compactness theorems with new notions of convergence are developed in this project providing fundamental new geometric tools that can be applied to address these challenges. The principal investigator has already been invited to present preliminary work in this direction at various mathematics and physics institutions around the world. As she has in the past, the PI will include young mathematicians of diverse backgrounds in this research project.The PI will seek intrinsic flat limits of noncollapsing sequences of Riemannian manifolds with uniform lower bounds on scalar curvature. For example, the PI will consider sequences of asymptotically flat Riemannian manifolds with nonnegative scalar curvature whose ADM mass is approaching zero, or regions in such spaces with a uniform upper bound on Hawking mass. Compactness theorems for such sequences would be useful to prove the Almost Rigidity of the Schoen-Yau Positive Mass Theorem or the Bartnik Conjecture. Similar methods will also be applied towards proving Gromov's Almost Rigidity of Flat Tori Conjecture. To avoid cancellation and bubbling, the PI proposes to forbid the existence of arbitrarily small closed minimal surfaces in these and other conjectures stated within the proposal. Various Compactness Theorems for Intrinsic Flat convergence have been proven in different settings by Prof. Wenger, Dr. Portegies, Prof. LeFloch, Dr. Perales, Dr. Matveev, and the PI. Prior applications of intrinsic flat convergence to General Relativity have been completed in various papers by Prof. Lee, Prof. Huang, Prof. LeFloch, Prof. Stavrov, Prof. Jauregui and the PI.