Mathematical Sciences: On Some Geometric Constructions and On the Properties of the Kerr Black Hole

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9704338 Kapouleas The proposal consists of two somewhat distinct groups of projects. The first group of project deals with various geometric constructions by singular pertubation methods. These include constructions using desingularization of intersecting minimal surfaces, minimal hypersurfaces and constant mean curvature hypersurfaces in higher dimensional Euclidean space, and Einstein four-manifolds. The second group of projects lie in the area of general relativity. In particular, the PI, in collaboration with Christodoulou, will try to settle uniqueness and stability questions for the Kerr and Schwarzschild black holes. Minimal surfaces arise as various physical surfaces; they have the remarkable property of minimizing the surface area amongst all surfaces bounded by the same spatial contour. Minimal surfaces can also used to model condensed matter. Stability questions for black holes, which involve examining the interaction of a black hole with weak gravitational waves, have important applications in cosmology.