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Some remarks on compact constant mean curvature hypersurfaces in a halfspace of $$\mathbb{H}^{n+1}$$. (English) Zbl 0981.53049
The authors give a theorem (see the abstract of the paper for a complete statement of it) for hypersurfaces of constant mean curvature in a halfspace of hyperbolic space $$\mathbb H ^{n+1}$$. They consider embedded compact hypersurfaces $$M$$ with boundary $$\partial M$$ in the boundary geodesic hyperplane $$P$$ of the halfspace and with non-zero mean curvature. They also prove a result about the topology of such hypersurfaces.

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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References:
 [1] Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V. Vestnik Leningrad Univ., 13 No. 19 A.M.S. (Series 2)21 (1958) 412-416. [2] De Miranda Gomez, J.: Sobre hipersuperficies com curvatura media constante no espaco hiperbolico. PhD thesis, IMPA (1985). [3] Gilbarg, D. andTrudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag (1983). · Zbl 0562.35001 [4] Korevaar, N. andKusner, R. andMeeks III,W.H. andSolomon, B.: Constant mean curvature surfaces in hyperbolic space. American Journal of Mathematics114 (1992) 1-43. · Zbl 0757.53032 · doi:10.2307/2374738 [5] Ros, A. andRosenberg, H.: Constant mean curvature surfaces in a half-space of ?3 with boundary in the boundary of the half-space. To appear in Journal of Differential Geometry (1997). [6] Semmler, B.: Blow up theorems for compact constant mean curvature surfaces. Preprint (1997).
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