Constant mean curvature surfaces in Euclidean and Minkowski three-spaces. (English) Zbl 1183.53005

Brander, David; Rossman, Wayne; Schmitt, Nicholas

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 10th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–11, 2008. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-323-531-5/pbk). 133-142 (2009).

From the authors’ abstract: Space-like constant mean curvature (CMC) surfaces in Minkowski three-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 \(\text{SU}(2)\) with \(\text{SU}(1,1)\). 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.
Reprint of J. Geom. Symmetry Phys. 12, 15–26 (2008; Zbl 1159.53336).
For the entire collection see [Zbl 1169.81004].

Reviewer: Ivan C. Sterling (St. Mary’s City)

MSC:

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

Keywords:

constant mean curvature; Minkowski space; generalized Weierstrass representation; Iwasawa decomposition; loop group

Citations:

Zbl 1159.53336