Morse Index Bounds and Degeneration of Surfaces and Manifolds

对于DMS-0104453，这个项目的目标是描述固定(但任意)闭三维流形中的所有固定(但任意)亏格的嵌入极小曲面。这样的研究将包括在固定的小尺度上描述这些表面。还将研究这种描述的各种应用。特别是球面空间型问题、具有正标量曲率的3-流形的拓扑以及3-流形拓扑中的其他各种问题。另外两个单独的项目将被调查。一种是曲面上简单的闭测地线，另一种是Kahler-Einstein流形的退化。粗略地说，这个提议的目标是描述所有光滑的肥皂膜。肥皂膜是一个跨越边界的曲面，对于具有相同边界的所有其他曲面的面积来说，它是至关重要的。这种描述除了是一个经典问题外，还可能在广泛的领域中有许多可能的应用。从拓扑学到数学生物学。

Abstract for DMS - 0104453The goal of this project is to describe all embedded minimal surfaces of some fixed (but arbitrary) genus in a fixed (but arbitrary) closed 3-manifold. Such a study would include the description of these surfaces on a fixed small scale. Various applications of such a description will also be studied. In particular to the spherical space-form problem, the topology of 3-manifolds with positive scalar curvature and various other problems in the topology of 3-manifolds. Two other seperate projects will be investigated. One is simple closed geodesics on surfaces and another is degeneration of Kahler-Einstein manifolds.Roughly speaking the goal of this proposal is to describe all smooth soap-films. I soap-film is a surface spanning a boundary and which is critical for area among all other surfaces with the same boundary. Besides being a classical problem such a description would have many possible applications in a wide varieties of fields. Spanning from topology to mathematical biology.