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Period problems for mean curvature one surfaces in \(H^3\) (with applications to surfaces of low total curvature). (English) Zbl 1171.53041

Guest, Martin (ed.) et al., Surveys on geometry and integrable systems. Based on the conference on integrable systems in differential geometry, Tokyo, Japan, July 17–21, 2000. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-46-4/hbk). Advanced Studies in Pure Mathematics 51, 347-399 (2008).
The authors investigate the existence of complete surfaces in hyperbolic 3-space where the mean curvature of these surfaces is constant and equal to one (i. e. CMC-1 surfaces). Constructing examples of complete CMC-1 surfaces the authors have to solve period problems on non-simply-connected domains of the immersions. In this way, the authors present many examples of complete CMC-1 surfaces for which the total absolute curvature does not exceed \( 8\pi \) (see also the papers of the authors in [Hiroshima Math. J. 34, No. 1, 21–56 (2004; Zbl 1088.53004) and Tohoku Math. J., II. Ser. 55, No. 3, 375–395 (2003; Zbl 1058.53008)]). On the other hand, in some cases which depend on the type of the surface they disprove the existence of such surfaces. Furthermore, the question of existence is still open for other types. Finally, some figures of further examples of CMC-1 surfaces are given.
For the entire collection see [Zbl 1153.53003].

MSC:

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