Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods

TL;DR

This paper develops a generalized Weierstrass representation for spacelike constant mean curvature surfaces in Minkowski space, addressing non-compactness issues in loop group methods and classifying various symmetric surfaces.

Contribution

It introduces an infinite-dimensional representation for CMC surfaces in Minkowski space, overcoming non-global Iwasawa decomposition challenges and classifying symmetric examples.

Findings

Established a partial Iwasawa decomposition for the non-compact loop group

Constructed new examples of spacelike CMC surfaces with symmetries

Classified surfaces of revolution and screw motion symmetric surfaces

Abstract

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form $SU_{1,1}$, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in $\real^{2,1}$. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.