Teichmuller Theory and Low-Dimensional Geometric Variational Problems

William Marsh Rice University, Houston TX

Investigators

Abstract

Abstract Award: DMS-0505603 Principal Investigator: Michael Wolf These projects concentrate on problems concerning complete minimal surfaces in Euclidean three-space, harmonic maps onto the hyperbolic plane, and projective structures on Riemann surfaces. The principal investigator, working with D. Hoffman and M. Weber, hopes to prove the existence of embedded helicoids of arbitrary genus, following up on their existence result in genus one but pursuing a different approach that is already known to simplify the genus-one argument. A basic uniqueness question on minimal surfaces is also going to be studied: In how many ways can one desingularize the intersection of two planes? The principal investigator has studied harmonic maps onto the hyperbolic plane for several years, leading to a natural conjecture on extensions of quasi-symmetries of the circle to the hyperbolic disk. The planned work on projective structures on Riemann surfaces will be pursued jointly with a postdoctoral fellow, David Dumas, and aims to develop fine asymptotics for such structures. A minimal surface is the mathematical idealization of a soap bubble spanning a wire. A stable soap bubble ordinarily assumes the least area of all possible surfaces spanning that wire, and the mathematical statement of the fact that varying the surface must increase its area translates into a partial differential equation. Versions of these problems have been studied intensely since the 19th century and before, for both physical and geometric reasons. The helicoid referred to above is a surface shaped like a corkscrew or parking ramp, and the genus-one version of the helicoid could be described as a parking ramp with an airshaft -- some interesting and appealing pictures of these and other minimal surfaces are available on M. Weber's Web pages, http://www.indiana/edu/~minimal.