Mathematical Sciences: Applications of Variational Problems on Riemann Surfaces to Low Dimensional Geometry

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9626565 Wolf The proposed research lies in the area of variational calculus and minimal surfaces. Specifically, two topics are proposed: Teichmuller theoretic methods for finding minimal surfaces in Euclidean 3-space, and the geometry of a hyperbolic three-manifold via families of harmonic maps of surfaces thereto. Minimal surfaces first arose in a series of soap film experiments done by the Belgian physicist Joseph Plateau in the 19th century. These surfaces have an area-minimizing property: given a closed contour in space, the minimal surface spanning it has the smallest surface area amongst all surfaces bounding the same contour. Because of this and other extremal properties minimal surfaces find many applications in physics, chemistry and materials research. The proposed research has to do with constructing families of such minimal surfaces using complex analysis methods and a study of negatively curved 3-spaces using minimal surface techniques.