Mathematical Sciences: Applications of Minimal Surfaces to Curvature and Topology of Riemannian Manifolds and Submanifolds

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The principal investigator will develop a Morse theory of minimal two-spheres analogous to that of closed geodesics. A goal of the project is to construct large numbers of minimal two- spheres of low area using a result of the investigator concerning Morse theory at low energies. The investigator will also exploit the relationship between minimal two-spheres in Riemannian manifolds and holomorphic vector bundles over the Riemann sphere. And an extension of Hilbert's theorem to n-dimensional submanifolds of constant negative curvature in odd-dimensional Euclidean space will also be investigated. This goal of this project is to understand minimal surfaces with low area. Soap bubbles are examples of minimal surfaces which minimize energy. The principal investigator will use Morse theory to study these surfaces and their generalizations to higher dimensions. This theory is an extension or our concept of elevation, where basins and mountain passes undergo separate mathematical interpretations.