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

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During the last decade Riemannian geometry has undergone a revolution following the introducton of new techniques from the theory of minimal surfaces. In recent joint work with M.Micallef, the principal investigator has been at the forefront of these developments. In this work they strengthened the Klingenberg- Berger sphere theorem. This involved use of Morse theory for minimal two-spheres. John Moore proposes to continue applying these innovative techniques to find further relationships between curvature and topology for Riemannian manifolds. He will also be studying submanifolds of low codimension in Euclidean space. The proposed problems are both interesting and difficult. Not only would their solution be of significance, but the development of these new techniques is of great importance.