Elliptic and Parabolic Problems in General Relativity

摘要DMS -0205545.本文将研究广义相对论中的一些椭圆型和抛物型问题。 该模型采用摄动方法，将自引力理想流体作自转运动模拟旋转恒星。扰动参数是角速度与中心密度的比值，当密度足够高时，允许快速旋转的解决方案。研究人员将致力于克尔黑洞的独特性，以排除多个黑洞配置，如罗杰彭罗斯所示。 本文提出了一种新的研究方法，即研究平衡态黑洞的唯一性定理与Penrose型不等式之间的关系。研究人员将继续他的工作的空间的连通性的有限数据的爱因斯坦真空方程，扩展的结果中获得的上下文中的准凸叶理，叶理与叶子的正高斯和平均曲率。 该系统将与Y. Li和A. S.Tahvildar-Zadeh，关于波映射的爆破，调和映射的动力学对应物。 这些映射由于与爱因斯坦方程的相似性而引起了非线性双曲方程研究者的广泛关注。 研究者还将致力于研究具有指定奇点的调和映射到Hadamard流形的渐近行为。数学广义相对论重叠了三个学科：物理学、几何学和非线性偏微分方程。因此，它是一个理想的问题源，其解决方案有可能影响所有这三个领域。 这个项目中的问题属于这一类。 例如，彭罗斯型不等式的研究是数学相对主义者和研究三维空间的几何学家都感兴趣的。发展一个好的致密快速旋转恒星的数学模型是天体物理学的一个重要目标，例如在开始理解恒星时，也将在微扰理论中贡献新的结果。 理解波映射有助于研究其他控制各种物理现象的非线性演化方程。

ABSTRACT DMS - 0205545.A number of problems of elliptic and parabolic type ingeneral relativity will be studied. The investigatorwill model rotating stars by a self-gravitating perfect fluid rigidly rotating on its own axis using aperturbation method. The perturbation parameter is theratio of the angular velocity to the central density,allowing rapidly rotating solutions when the densityis sufficiently high. The investigator will work onthe uniqueness of the Kerr black holes, to rule outmultiple black hole configurations, as conjectured byRoger Penrose. A new approach is proposed consistingof the investigation of the relation betweenuniqueness theorems for black holes in equilibrium andPenrose-type inequalities. The investigator willcontinue his work on the connectedness of the space ofinitial data for the Einstein vacuum equations,extending results obtained in the context ofquasi-convex foliations, foliations with leaves ofpositive Gauss and mean curvature. The investigatorwill work, in collaboration with Y. Li and A. S.Tahvildar-Zadeh, on blow-up of wave maps, thedynamical counterparts of harmonic maps. These mapshave attracted considerable attention by researchersin nonlinear hyperbolic equations mainly due to theirsimilarity to the Einstein equations. Theinvestigator will also work on the asymptotic behaviorof harmonic maps with prescribed singularities intoHadamard manifolds.Mathematical general relativity overlaps threedisciplines: physics, geometry, and nonlinear partialdifferential equations. As such, it is an ideal sourceof problems whose solutions have the potential toinfluence all three of these fields. The problems inthis project fall in this category. For example, thestudy of Penrose-type inequalities is of interest bothto mathematical relativists and to geometers studyingspaces of dimension three. Developing a goodmathematical model for dense rapidly rotating stars isan important goal in astrophysics, for example whentrying to understand pulsars, and would alsocontribute new results in perturbation theory. Understanding wave maps should help the study of othernonlinear evolution equations governing variousphysical phenomena.