Teichmuller theory and Low-Dimensional Geometric Variational Problems

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AbstractAward: DMS-1007383Principal Investigator: Michael WolfThe principal investigator will continue his research in two-dimensional geometric variational problems. In particular, he proposes projects in several areas of classical minimal surface theory, all concerned with using Teichmuller theory to prove existence or uniqueness or properties of minimal surfaces in Euclidean three-space. He proposes research in two areas of Teichmuller theory: Weil-Petersson geometry and grafting. Both bring a Riemannian perspective to problems often studied via other methods. In addition, he proposes studying problems arising in convex projective structures on surfaces as well as work on degenerating harmonic quasi-isometries of the hyperbolic plane. These last are informed by his earlier work on high energy harmonic maps. In recent years, the PI has focused considerable attention upon education at all levels. He supervises graduate students; he coordinates a VIGRE program at Rice University encompassing roughly a dozen small research groups in the mathematical sciences whose vertical organization ranges from the undergraduate stratum through the graduate and postdoctoral level to the permanent faculty; he participates in one of these groups himself, he lectures to teachers and acts as a scientific advisor to programs to enhance the mathematical understanding of K-12 teachers. He serves the broader mathematical community by serving on the editorial board of three journals, through organizing large and small conferences and through lectures to general audiences. He is quite involved with undergraduate advising. The PI will continue with all of these activities.One of the guiding principles of science is that Nature is efficient: when we encounter natural phenomenon, we expect that the shapes we find will use the least material for their construction, or be the thinnest that can be made with a material or will span the largest region possible, or some other sort of optimization. At the same time, a common geometric component of many natural systems is a two-dimensional surface, usually configured in some curved manner. These surfaces occur at all scales, from the boundary of a protein to the surface of the brain to the frontier of the magnetosphere. In this project, we investigate problems that relate aspects of the shapes of surfaces to quantities they might optimize, and also how deforming one feature of the shape of a surface affects other qualities of that surface.