DIFFERENTIAL GEOMETRY/PDE SEMINAR

Wednesday, October 18 Padelford C-36 3:45 PM

Minimal surfaces, conformal mappings, and Teichmüller theory

Matthias Weber (Mathematisches Institut Universität Bonn and MSRI)

As is well known, minimal surfaces in euclidean space are closely related to conformal geometry in several ways. For instance, the Gauß map of a minimal surface is conformal, and there are other interesting conformal maps associated to it. In fact, many geometric properties of a minimal surface can be translated into poperties of the conformal maps, using classical techniques as the Schwarz-Christoffel formula. Likewise, existence problems for minimal surfaces can sometimes be stated purely in terms of conformal geometry. Then, to solve these reformulated problems, methods from Teichmüller theory are used.

I will report about recent joint work with Michael Wolf (Rice University, Houston) on the construction of minmal surfaces of arbitrarily high genus generalizing the Costa-Hoffman-Meeks surface and on the embeddedness of the Genus One Helicoid.