Elliptic and Parabolic Problems in General Relativity

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

ABSTRACT DMS - 0205545. A number of problems of elliptic and parabolic type in general relativity will be studied. The investigator will model rotating stars by a self-gravitating perfect fluid rigidly rotating on its own axis using a perturbation method. The perturbation parameter is the ratio of the angular velocity to the central density, allowing rapidly rotating solutions when the density is sufficiently high. The investigator will work on the uniqueness of the Kerr black holes, to rule out multiple black hole configurations, as conjectured by Roger Penrose. A new approach is proposed consisting of the investigation of the relation between uniqueness theorems for black holes in equilibrium and Penrose-type inequalities. The investigator will continue his work on the connectedness of the space of initial data for the Einstein vacuum equations, extending results obtained in the context of quasi-convex foliations, foliations with leaves of positive Gauss and mean curvature. The investigator will work, in collaboration with Y. Li and A. S. Tahvildar-Zadeh, on blow-up of wave maps, the dynamical counterparts of harmonic maps. These maps have attracted considerable attention by researchers in nonlinear hyperbolic equations mainly due to their similarity to the Einstein equations. The investigator will also work on the asymptotic behavior of harmonic maps with prescribed singularities into Hadamard manifolds. Mathematical general relativity overlaps three disciplines: physics, geometry, and nonlinear partial differential equations. As such, it is an ideal source of problems whose solutions have the potential to influence all three of these fields. The problems in this project fall in this category. For example, the study of Penrose-type inequalities is of interest both to mathematical relativists and to geometers studying spaces of dimension three. Developing a good mathematical model for dense rapidly rotating stars is an important goal in astrophysics, for example when trying to understand pulsars, and would also contribute new results in perturbation theory. Understanding wave maps should help the study of other nonlinear evolution equations governing various physical phenomena.