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

9704338 Kapouleas提案由两组不同的项目组成。第一组项目用奇异摄动方法处理各种几何结构。这些包括使用相交最小曲面的去广化构造，高维欧几里德空间中的最小超曲面和常平均曲率超曲面，以及爱因斯坦四流形。第二组项目在广义相对论领域。特别是，PI将与Christodoulou合作，试图解决克尔黑洞和史瓦西黑洞的唯一性和稳定性问题。最小表面作为各种物理表面出现；它们具有在由同一空间轮廓所包围的所有表面之间最小化表面积的显著特性。最小表面也可以用来模拟凝聚态物质。黑洞的稳定性问题，包括检查黑洞与弱引力波的相互作用，在宇宙学中有重要的应用。

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.