Mathematical Sciences: Research in Differential Geometry and Topology

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9505101 Meeks A variety of problems in differential geometry and topology of minimal and constant mean curvature surfaces are proposed. Total Gaussian curvature results as well as topological obstruction results will be sought for complete minimal surfaces of finite total curvature. For infinite total curvature minimal surfaces, topological uniqueness, growth estimates and the index theory will be examined. Global properties of constant mean curvature surfaces will be studied. Computer graphics and (period) computations for some of these surfaces will also be pursued. Minimal surfaces and constant mean curvature surfaces were first systematically investigated by the Belgian physicist Plateau in his famous soap film experiments. These surfaces model physical surfaces where either the surface area is minimized by surface tension or in the case of a stable interface between two materials. In recent years, periodic minimal surfaces are used to model condensed matter on a molecular scale.