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

9704338卡普里亚该提案由两组不同的项目组成。第一组项目用奇异摄动法处理各种几何结构。其中包括利用高维欧氏空间中相交极小曲面、极小超曲面和常平均曲率超曲面的去奇化构造，以及爱因斯坦四维流形。第二组项目位于广义相对论领域。特别是，PI将与Christodoulou合作，试图解决Kerr和Schwarzschild黑洞的唯一性和稳定性问题。极小曲面是以各种物理曲面的形式出现的，它们具有使同一空间等高线所包围的所有曲面的表面积最小的显著性质。极小曲面也可以用来模拟凝聚态物质。黑洞的稳定性问题，包括研究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.