RUI: Problems in Geometric Analysis and General Relativity

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。侦探他的项目适合于理解约束方程解的空间的广泛计划，通常具有指定的渐近性。P.I.开发了局部变形和渐近粘接技术，这些技术在理解约束解的渐近结构和在约束解上执行局部粘接构造方面都被证明是富有成效的。例如，P.I.与r.s Schoen的联合工作表明，具有完全Kerr渐近的真空约束的解在渐近平坦解的空间中是密集的。项目的一部分涉及研究渐近结果到渐近双曲设置的推广，这将在这种情况下应用于正质量定理（继Andersson-Cai-Galloway最近的工作）。关于渐近性和局部胶合结构的结果将推广到重力与物质场耦合的情况。该项目的相关方面包括：约束的小数据解的研究（例如，与Penrose紧化有关）；结合准局部质量极小值的约束的局部变形性质，在Bartnik意义上；渐近平坦空间中视界和等周剖面的存在性洛伦兹比较几何中的几个问题。更透彻地理解爱因斯坦约束方程的解空间，自然会更好地理解爱因斯坦场方程的解空间，后者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，人们可以构造具有所需性质的初始数据，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数值模拟。自然物理问题往往变成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作业、独立学习和研究，向本科生介绍和吸引他们了解物理和几何分析的奇妙相互作用。学生将在夏季与PI一起进行研究（与拉斐特大学的REU项目同时进行，因此可以进行更广泛的讨论和分享结果），并在一年中作为独立学习和荣誉论文工作。

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.