Constant Mean Curvature Surfaces in Euclidean and Minkowski Three-Spaces

Abstract

Spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\mathbb{L}^3$ have an infinite dimensional generalized Weierstrass representation. This is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $\mathrm{SU}(2)$ with $\mathrm{SU}(1,1)$. 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. The construction is described here, with an emphasis on the difference from the Euclidean case.