Geometric Compactness Theorems with Applications to General Relativity

在广义相对论中，时空和时空的类空间片段是流形（局部类似于欧几里得空间的物体），满足由爱因斯坦方程和其他物理自然约束决定的某些几何条件。广义相对论中出现的流形因重力而弯曲，它们可能包含黑洞或薄而深的重力井，这使得从技术上难以估计流形与简化模型（如欧几里得空间）的接近程度。本项目开发了新的紧性定理和新的收敛概念，为解决这些挑战提供了新的基本几何工具。首席研究员已经被邀请到世界各地的数学和物理机构展示这个方向的初步工作。和过去一样，PI将包括不同背景的年轻数学家参与这个研究项目。PI将寻求在标量曲率上具有一致下界的黎曼流形的非坍缩序列的内在平极限。例如，PI将考虑具有非负标量曲率的渐近平坦黎曼流形序列，其ADM质量接近于零，或者这样的空间中具有均匀霍金质量上界的区域。这类序列的紧性定理将有助于证明Schoen-Yau正质量定理或Bartnik猜想的几乎刚性。类似的方法也将用于证明Gromov的平环面猜想的几乎刚性。为了避免消去和冒泡，PI建议在这些和建议中所述的其他猜想中禁止任意小的封闭最小曲面的存在。Wenger教授、Portegies博士、LeFloch教授、Perales博士、Matveev博士和PI已经在不同的环境下证明了内在平坦收敛的各种紧性定理。在此之前，Lee教授、Huang教授、LeFloch教授、Stavrov教授、Jauregui教授和PI已经在不同的论文中完成了广义相对论内禀平面收敛的应用。

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.