Geometric Compactness Theorems with Applications to General Relativity

在广义相对论中，时空和时空的类空切片是流形（局部类似于欧几里得空间的物体），满足爱因斯坦方程和其他物理自然约束所确定的某些几何条件。在广义相对论中出现的流形被引力弯曲，它们可能包含黑洞或薄的深重力威尔斯井，这使得在技术上很难估计流形与简化模型（如欧几里得空间）的接近程度。 在这个项目中开发了新的收敛概念的新紧性定理，提供了基本的新几何工具，可以应用于解决这些挑战。首席研究员已应邀在世界各地的数学和物理研究所介绍这方面的初步工作。 正如她过去所做的那样，PI将在这个研究项目中包括不同背景的年轻数学家。PI将寻求黎曼流形的非塌缩序列的内在平坦极限，其标量曲率具有一致的下界。例如，PI将考虑具有非负标量曲率的渐近平坦黎曼流形的序列，其ADM质量接近零，或者在这样的空间中具有霍金质量的一致上界的区域。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。类似的方法也将被应用于证明Gromov的几乎刚性平坦环面猜想。为了避免取消和冒泡，PI建议禁止在提案中陈述的这些和其他猜想中存在任意小的闭合极小曲面。Wenger教授、Portegies博士、LeFloch教授、Perales博士、Matveev博士和PI在不同的环境中证明了各种内在平坦收敛的紧性定理。内在平坦收敛在广义相对论中的应用已经在Lee教授、Huang教授、LeFloch教授、Stavrov教授、Jauregui教授和PI的多篇论文中完成。

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.