Abstract of Proposed Research Sergiu Klainerman
The research focuses on the mathematical analysis of the evolution of the solutions of the Einstein-field equations. Attention will be concentrated on a number of conjectures about these solutions. One concerns the relationship between the curvature tensor and the causal geometry of space-time. Two concern the phenomenon of "cosmic censorship", other concerns the asymptotic behavior or "final state" and one about the stability of the Kerr solution. The resolution of these problems will require the study of the relationship between geometry and curvature of Lorentzian manifolds as well as analysis of the Einstein equations.
Einstein's equations are the field equations of general relativity and their analysis is central to the understanding of many issues in cosmology including the theory of black holes and gravitational waves. There are only a few exact solutions known of these equations and this research will pursue a more mathematical analysis that will prove qualitative results about the solutions. This analysis should also help the development of algorithms for the numerical computation of solutions of these equations.
The focus of research for the NSF grant 0601186 was the theory of Black Holes. These are not only specific stationary (i.e., roughly, not varying with time) of the Einstein field equations but also real astrophysical objects of great importance to our understanding of Nature. Roughly, they correspond to regions of space of intense gravitational forces from which no physical signals can escape. I have concentrated my research in two important problems related to these solutions: uniqueness and stability. I have also worked on mathematical problems connected to the cosmic censorship conjecture, due to Penrose, which concern the structure of general solutions (i.e. not just stationary) of the Einstein field equations. At the heart of the problem of uniqueness is a precise conjecture which affirms that all stationary solutions of the Einstein field equations (in vacuum) must belong to a specific class of solutions discovered by the applied mathematician Kerr fifty years ago. A well known result of Carter, Robinson and Hawking solves this important conjecture in the class of real analytic spacetimes. The assumption of real analyticity is however very problematic; there is simply no physical or mathematical justification for it. Six or seven years ago A. Ionescu and I have initiated a program to dispense of the analyticity by relying instead on a geometric version of the method of unique continuation. We have obtained a sequence of results the most important of which proves the desired uniqueness (without analyticity) for stationary solutions which are reasonable close to a given Kerr solution. I have obtained other significant results in collaboration with I. Rodnianski on ``The breakdown criterion criterion in GR'' and with I. Rodnianski and J. Szeftel `` The bounded $L2$ curvature conjecture '' Both results are important advances in the quest of understanding the properties of general solutions of the Einstein field equations. They both give precise criteria about the possible singularities of spacetimes, i.e. solutions to the field equations.
