On the Behavior of Solutions of Einstein's Equations and Other Geometric Partial Differential Equation Systems

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The research supported by this award covers a wide variety of projects in general relativity, gravitational physics, and geometric analysis. A number of the projects are concerned with what Einstein's equations for the gravitational field tell us about properties of black holes. We hope to complete a proof that, in any spacetime dimension, black holes in equilibrium must have spatial symmetries. We also plan to continue a program of study of model astrophysical systems which are not in equilibrium: when can one identify such a system as a future black hole, and can one do this in a topologically closed universe? Other projects we are proposing involve mathematically probing the nature of gravitational fields near the Big Bang: is a curvature (tidal) singularity generally to be found near the Big Bang, and does the current support for oscillatory behavior seen in restricted families of cosmological models in fact extend to less restricted families?In addition to these projects involving general relativity, our proposed research includes studies of Ricci flow. This system of partial differential equations, which has recently played a major role in the proof of the Poincare and the geometrization conjectures, is a promising tool for further studies of the relationship between geometry and topology. We have had success in studying the stability of certain equilibrium geometries in Ricci flow, and we plan to extend this work to a wider class (including collapsing geometries.)