Elliptic and Parabolic Problems in General Relativity

摘要DMS-0205545将研究广义相对论中椭圆和抛物型的若干问题。研究人员将使用微扰方法，通过自重的完美流体刚性地绕自己的轴旋转来模拟旋转的恒星。微扰参数是角速度与中心密度的比值，当密度足够高时，允许快速旋转解。研究人员将致力于研究克尔黑洞的独特性，以排除罗杰·彭罗斯猜测的多种黑洞配置。提出了一种研究平衡态黑洞唯一性定理与Penrose型不等式之间关系的新方法。研究者将继续他在爱因斯坦真空方程的初始数据空间的连通性方面的工作，推广在拟凸叶、具有正高斯和平均曲率的叶的背景下所获得的结果。这位研究人员将与Y.Li和A.S.Tahvildar-Zadeh合作，研究波映射的爆破，即调和映射的动力学对应。这些映射引起了非线性双曲型方程研究人员的极大关注，这主要是因为它们与爱因斯坦方程的相似性。研究人员还将研究Hadamard流形上具有指定奇点的调和映射的渐近行为。数学广义相对论涉及三个学科：物理、几何和非线性偏微分方程式。因此，它是一个理想的问题来源，其解决方案有可能影响这三个领域。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对数学相对主义者和研究三维空间的几何学家都很感兴趣。为密集的快速自转恒星建立一个良好的数学模型是天体物理学的一个重要目标，例如理解脉冲星，也将为微扰理论贡献新的成果。了解波图应该有助于研究控制各种物理现象的其他非线性演化方程。

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.