Recent Work on Integrable Surfaces Via the Generalized Weierstrass Representation
David Brander
Technical University of Denmark
Abstract
the classical Weierstrass representation for minimal surfaces has been an important tool in their study, both for constructing examples and for proving general properties about these surfaces. For non-minimal CMC surfaces, a generalized Weierstrass representation was given by Dorfmeister, Pedit and Wu in the 1990's. Analogously to the classical Weierstrass representation for minimal surfaces, all such surfaces can be represented by pairs of holomorphic functions. This representation has been used to construct new examples of CMC surfaces, such as CMC trinoids and generalizationis, but, in general the method is much less straightforward to make use of compared with the minimal case. This is due to a loop group splitting which must be performed in the passage from the Weierstrass data to the surface, which makes it hard to detect geometric properties of the surface in the holomorphic data. In this talk, I will talk about some recent work on solving the generalization of Bjorling's problem to the non-minimal case via the loop group representation. Other generalizations, which will be mentioned if time permits, include the analogous problem for constant negative curvature surfaces and also the construction of CMC surfaces in Lorentz 3-space with prescribed singularities.