Nonlinear Partial Differential Equations in Relativity and Geometry

这个项目将解决一些问题，主要在以下两个相关领域：调和映射和爱因斯坦方程。研究者将研究调和映射在指定奇点附近的负曲空间中的渐近行为。这项研究的动机是广义相对论中的静止黑洞问题，并适用于此问题。PI还建议推广波图的爆破结果。由规则的初始数据产生奇异性的波图将由定义在双曲空间上的调和映射构造。在广义相对论领域，研究人员将继续研究同轴旋转的带电黑洞的平衡组态。寻求与存在的黑洞数量无关的唯一性定理。还提出了对爱因斯坦方程初始数据空间的拓扑结构的研究，以及与旋转和非旋转理想流体的平衡构型有关的问题。调和映射是满足非线性椭圆型偏微分方程的黎曼流形(几何空间)之间的映射。它们首先是由C.B.Morrey Jr.在二维域上引入的。20世纪50年代，S在他的黎曼流形上的高原问题的研究中：寻找一种跨越给定轮廓的肥皂片。自那以后，调和映射在物理、几何和其他领域得到了大量的应用。在这个项目中，具有指定奇点的调和映射将被用来模拟同轴旋转的带电黑洞的平衡构型。了解这些构型是未来研究演化方程的重要一步。波映射是调和映射的动力学对应。它们满足非线性双曲型偏微分方程组。这些方程已经被许多研究人员作为理解更复杂的经典场论演化方程的程序的第一步，例如广义相对论的爱因斯坦场方程。爱因斯坦场方程组模拟了时空的大尺度演化。这类非线性发展方程的一个有趣的特征是，从规则数据开始的解有时可能在有限时间圈后产生爆破，从而导致奇点。研究人员将构建和研究具有这种奇点的波图。这个项目要解决的另一个应用于天体物理学的问题是，为处于平衡状态的相对论致密旋转恒星构建一个模型。

This project will address a number of problems, mainly in the following two related areas: harmonic maps, and the Einstein equations. The investigator will study the asymptotic behavior of harmonic maps into negatively curved spaces near prescribed singularities. This study is motivated by, and has applications to, the problem of stationary black holes in general relativity. The PI also proposes to generalize blow-up results for wave maps. Wave maps which develop singularities from regular initial data will be constructed from harmonic maps defined on hyperbolic space. In the area of general relativity, the investigator will continue his work on equilibrium configurations of co-axially rotating charged black holes. A uniqueness theorem is sought independent of the number of black holes present. Also proposed is the study of the topology of the space of initial data for the Einstein equations, and questions related to equilibrium configurations of rotating and nonrotating perfect fluid bodies. Harmonic maps are mappings between Riemannian manifolds (geometrical spaces) which satisfy nonlinear elliptic partial differential equations. They were first introduced on two dimensional domains by C. B. Morrey Jr. in the 1950's in his study of Plateau's Problem in Riemannian manifolds: to find a soap film which spans a given contour. Harmonic maps have since found numerous applications in physics, geometry, and other fields. Harmonic maps with prescribed singularities will be used in this project to model equilibrium configurations of co-axially rotating charged black holes. Understanding these configurations is an important step in the future study of the evolution equations. Wave maps are the dynamical counterparts of harmonic maps. They satisfy nonlinear hyperbolic partial differential equations. These equations have been studied by many researchers as a first step in a program to understand more complex evolution equations of classical field theory such as the Einstein field equations of general rel ativity. The Einstein field equations model the large scale evolution of spacetime. One of the interesting features of this class of nonlinear evolution equations is that solutions starting from regular data may sometimes yield blow-up after a finite time laps, leading to singularities. The investigator will construct and study wave maps with such singularities. Another problem, with applications to astrophysics, to be addressed in this project will be to construct a model for relativistic dense rotating stars in equilibrium.