Geometric Compactness Theorems with Applications to General Relativity

几何紧性定理及其在广义相对论中的应用

• 批准号: 1612049
• 负责人: Christina Sormani
• 金额: $ 16.8万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612049&HistoricalAwards=false
• 关键词: Geometric; Compactness; Theorems; Applications; General

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.

在广义相对论中，时空和时空的类空切片是满足由爱因斯坦方程和其他物理自然约束确定的某些几何条件的流形(局部类似于欧几里得空间的对象)。广义相对论中产生的流形是由重力弯曲的，它们可能包含黑洞或细长的深重力井，因此在技术上很难估计流形与欧几里得空间等简化模型的距离。在这个项目中发展了新的紧致性定理和新的收敛概念，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各种数学和物理机构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将寻找具有一致标量曲率下界的黎曼流形的非折叠序列的内在平坦极限。例如，PI将考虑具有非负标量曲率且ADM质量趋近于零的渐近平坦黎曼流形序列，或此类空间中具有一致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。类似的方法也将被用来证明格罗莫夫平坦环面猜想的几乎刚性。为了避免抵消和冒泡，PI建议禁止在提案中所述的这些猜想和其他猜想中存在任意小的闭合极小曲面。Wenger教授、Portegie博士、LeFloch教授、Perales博士、Matveev博士和PI在不同的背景下证明了关于内在平坦收敛的各种紧性定理。李教授、黄教授、勒弗洛赫教授、斯塔夫罗夫教授、Jauregui教授和PI在不同的论文中完成了本征平坦收敛在广义相对论中的应用。