## 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].

### MSC:

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