Mathematical Sciences: The Global Geometry of Extremal Surfaces

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The principal investigator will study the global differential geometry of surfaces in three dimensional manifolds. The research focuses on surfaces which are extrema for natural symmetric variational problems arising in mathematics and physics. Computable geometric moduli for extremal surfaces will be developed. The robustness of this theory will be investigated by relaxing some symmetry assumptions on the ambient space. And these methods will be extended to other geometric variational problems with symmetry or approximate symmetry. Three-dimensional space is an example of a three-manifold. If infinity is collapsed to one point, this space becomes the three-dimensional sphere. This sphere has constant sectional curvature because relative to every plane at every point, the sphere bends the same. The principal investigator will study deep problems concerning the minimality of surfaces which lie in such three-manifolds. Soap films are familiar examples of such minimal surfaces.