Constant mean curvature surfaces in Euclidean and Minkowski three-spaces. (English) Zbl 1159.53336

Brander, David; Rossman, Wayne; Schmitt, Nicholas

J. Geom. Symmetry Phys. 12, 15-26 (2008).

Summary: 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 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. The construction is described here, with an emphasis on the difference from the Euclidean case.

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Keywords:

Weierstrass representation; Iwasawa decomposition; loop group; KDPW method; Birkhoff decomposition