A Plateau problem for complete surfaces in the de Sitter three-space. (English) Zbl 1152.53006

Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 156-168 (2007).
In this paper, the author establishes some existence and uniqueness theorems for a Plateau problem at infinity for complete space-like surfaces in the de Sitter 3-space \(S^3_1\) whose mean curvature \(H\) and Gauss curvature \(K\) verify the linear relationship \(2\varepsilon (H-1)-(\varepsilon +1)(K-1)-0\) for \(-\varepsilon\in R^{+}\). The Plateau problem considered in this paper is as follows: Given \(\varepsilon_0<0\) and a Jordan curve \(\Gamma \) on \(S^2_{\infty}\equiv\Pi\cup\{\infty\}\), find a complete BLW-surface \(\psi :S\to S^3_1\) verifying \(2\varepsilon_0(H-1)-(\varepsilon_0+1)(K-1)=0\) and such that \(\Gamma\) is its asymptotic boundary.
For the entire collection see [Zbl 1108.53003].


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A35 Non-Euclidean differential geometry