RUI: Problems in Geometric Analysis and General Relativity

P.I.将研究几何分析中的问题，这些问题既来自物理学(通过爱因斯坦方程)，也来自几何(几何变分问题和标量曲率)。爱因斯坦场方程的初始数据集满足一个非线性椭圆型方程组，即爱因斯坦约束方程。对这个系统的研究已经证明对几何分析是有趣的和富有成果的，并揭示了爱因斯坦场方程解的空间结构，这是物理学感兴趣的。在之前与R.Schoen的合作中，P.I.开发了约束算子的变形和粘合技术，(P.I.和其他人)已经使用这些技术来构造有趣的初始数据集，并导致对约束方程渐近平坦解的结构有了更好的理解。该项目的一部分涉及扩展关于渐近性和粘合的结果，包括建立初始数据模型N体构型，以包括某些物质模型，并分析在渐近平坦环境下发展的一些结果可以扩展到渐近双曲情况的程度。P.I.还将讨论关于约束的小数据解的结构以及局部标量曲率变形的应用的几个问题。在该项目的第二部分，P.I.将继续研究卡勒和辛几何中的一个变分问题，即哈密尔顿定常拉格朗日问题，这是在最近与A.Butscher的合作基础上进行的。另一个有趣的变分问题是等周问题(最小化封闭给定体积所需的面积)，它引出了关于密度流形和与广义相对论相关的空间的问题，适合本科生研究。爱因斯坦场方程的解被用来模拟引力辐射、黑洞等强场现象、孤立的引力系统和宇宙的大尺度结构。更详细地了解爱因斯坦约束方程的解空间将会更好地理解这些模型。例如，了解约束解的渐近结构可以提供有关引力辐射模型的信息。对约束系统的分析导致构建具有有趣属性的初始数据，例如其中两个或更多个孤立系统被融合在一起成为约束的连通解的解。对于引力辐射的研究来说，用特殊的渐近性数值实现小初始数据的构造，然后数值求解爱因斯坦场方程，这将是非常有趣的；类似的评论也适用于N体组态。该项目的一个重要方面是向本科生介绍几何、分析和物理之间的联系。学生将在暑假期间与PI一起进行研究(与Lafayette REU现场项目同时进行，进一步丰富了该系的暑期研究环境)，在学年期间，学生将与PI一起参与课程作业、独立学习和研究。

The P.I. will study questions in geometric analysis arising both from physics (via the Einstein equation) as well as from geometry (geometric variational problems and scalar curvature). Initial data sets for the Einstein field equation satisfy a nonlinear elliptic system of equations, the Einstein constraint equations. The study of this system has proven to be interesting and fruitful for geometric analysis, and has shed light on the structure of the space of solutions to the Einstein field equation, which is of interest for physics. In previous joint work with R. Schoen, the P.I. developed deformation and gluing techniques for the constraint operator which have been employed (by the P.I. and others) to construct interesting initial data sets, and have led to a better understanding of the structure of asymptotically flat solutions of the constraint equations. Part of the project involves extending results on asymptotics and gluing, including construction of initial data modeling N-body configurations, to include certain matter models, and analyzing the extent to which some of the results that have been developed in the asymptotically flat setting can extend to the asymptotically hyperbolic case. The P.I. will also address several questions on the structure of small-data solutions of the constraints, and as well as on applications of localized scalar curvature deformation. In a second part of the project, the P.I. will continue the study of a variational problem in Kahler and symplectic geometry, the Hamiltonian stationary Lagrangian problem, building on recent joint work with A. Butscher. Another interesting variational problem, the isoperimetric problem (minimizing the area required to enclose a given volume), leads to questions on manifolds-with-density and on spaces relevant to general relativity amenable to research with undergraduates. Solutions to the Einstein field equations are used to model gravitational radiation, strong field phenomena like black holes, isolated gravitational systems, and the large-scale structure of the universe. A more detailed understanding of the space of solutions to the Einstein constraint equations would yield a better understanding of these models. For instance, understanding the asymptotic structure of solutions to the constraints can yield information about models of gravitational radiation. Analysis of the constraint system leads to the construction of initial data with interesting properties, such as solutions in which two or more isolated systems are fused together into a connected solution of the constraints. It would be very interesting for the study of gravitational radiation to numerically implement constructions of small initial data with special asymptotics, and then numerically solve the Einstein field equations; similar comments apply to N-body configurations. An important aspect of the project is to introduce undergraduates to the connections between geometry, analysis and physics. Students will undertake research with the PI during the summer (occurring simultaneously with the Lafayette REU Site program, further enriching the summer research environment in the department), and during the academic year students will be involved in course work, independent study, and research with the PI.