RUI: Problems in Geometric Analysis and General Relativity

RUI：几何分析和广义相对论中的问题

• 批准号: 0707317
• 负责人: Justin Corvino
• 金额: --
• 依托单位: Lafayette College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707317&HistoricalAwards=false
• 关键词: RUI; Problems; Geometric; Analysis; General

Initial data for the Einstein field equations are specified by solutions to the Einstein constraint equations. The P.I.'s project fits into a broad program to understand the space of solutions to the constraint equations, often with specified asymptotics. The P.I. has developed local deformation and asymptotic gluing techniques that have proven fruitful, both for understanding the asymptotic structure of solutions of the constraints, and for performing localized gluing constructions on solutions to the constraints. For example, joint work of the P.I. with R. Schoen showed that solutions to the vacuum constraints which have exactly Kerr asymptotics are dense in the space of asymptotically flat solutions. Part of the project involves studying the extension of the asymptotic results to the asymptotically hyperbolic setting, which would have application to the Positive Mass Theorem in this case (following recent work of Andersson-Cai-Galloway). Results on both asymptotics and localized gluing constructions will be extended to the case of gravity coupled with matter fields. Related aspects of the project include: the study of small-data solutions of the constraints (relating to, for example, the Penrose compactification); local deformation properties of the constraints in conjunction with quasi-local mass minimizers, in the sense of Bartnik; the existence of horizons and isoperimetric profiles in asymptotically flat spaces; some problems in Lorentzian comparison geometry. A more thorough understanding of the space of solutions of the Einstein constraint equations would lead naturally to a better understanding of the space of solutions to Einstein's field equations, which are commonly used to model the large-scale structure of the universe. To understand the dynamics of gravitational systems, one might construct initial data with desired properties, like multiple horizons, or specified asymptotic structure, and then use such data for numerical modeling of the evolution into the resulting space-times. Natural physical questions often become interesting geometric and analytical problems involving the relevant partial differential equations and geometric quantities. A key component of the project is to introduce and attract undergraduates to this wonderful interplay of physics and geometric analysis, through course work, independent study and research. Students will undertake research with the PI during the summer (occurring simultaneously with Lafayette's REU program, thus, enabling a broader discussion and sharing of results) and during the year as independent study and honors theses work.

爱因斯坦场方程的初始数据由爱因斯坦约束方程的解指定。 私家侦探的项目适合于一个广泛的计划，以了解空间的解决方案的约束方程，往往与指定的渐近。 私家侦探已经开发了本地变形和渐近胶合技术，已被证明是富有成效的，无论是了解渐近结构的解决方案的约束，并进行本地化的胶合结构的解决方案的约束。 例如，P.I.与R. Schoen表明，解决方案的真空约束，这正是克尔渐近密集的空间中的渐近平坦的解决方案。 该项目的一部分涉及研究渐近结果的渐近双曲设置的扩展，这将在这种情况下应用于正质量定理（根据安德森-蔡-加洛韦最近的工作）。 渐近性和局部胶合结构的结果将扩展到重力与物质场耦合的情况。 该项目的相关方面包括：研究小数据的解决方案的约束（有关，例如，彭罗斯紧致化）;当地的变形性质的约束结合准当地的质量极小化，在巴特尼克的意义;存在的地平线和等周轮廓在渐近平坦的空间;在洛伦兹比较几何的一些问题。 对爱因斯坦约束方程的解的空间有更透彻的理解，自然会导致对爱因斯坦场方程的解的空间有更好的理解，爱因斯坦场方程通常用于模拟宇宙的大尺度结构。 为了理解引力系统的动力学，我们可以构造具有期望性质的初始数据，如多视界或指定的渐近结构，然后使用这些数据对演化到最终时空的过程进行数值建模。 自然物理问题往往成为有趣的几何和分析问题，涉及相关的偏微分方程和几何量。 该项目的一个关键组成部分是通过课程工作，独立学习和研究，介绍和吸引本科生物理和几何分析的这种奇妙的相互作用。 学生将在夏季与PI进行研究（与拉斐特的REU计划同时发生，因此，能够更广泛地讨论和分享结果），并在一年内作为独立研究和荣誉论文工作。