Morse Index Bounds and Degeneration of Surfaces and Manifolds

DMS -0104453的摘要本项目的目标是描述固定（但任意）闭3-流形中某些固定（但任意）亏格的所有嵌入极小曲面。 这种研究将包括在一个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。 特别是球面空间形式问题，具有正数量曲率的三维流形的拓扑以及三维流形拓扑中的其他各种问题。 将对另外两个项目进行调查。 一个是曲面上的简单闭测地线，另一个是Kahler-Einstein流形的退化，粗略地说，这个建议的目标是描述所有光滑的肥皂膜。 I皂膜是跨越边界的表面，并且在具有相同边界的所有其他表面中对于面积是关键的。 除了是一个经典的问题，这样的描述将有许多可能的应用在各种各样的领域。 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.