Morse Index Bounds and Degeneration of Surfaces and Manifolds

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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.