Nelli, Barbara; Rosenberg, Harold Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary. (English) Zbl 0831.53039 Ann. Global Anal. Geom. 13, No. 1, 23-30 (1995). Let \(M\) be a compact embedded hypersurface with boundary \(C = \partial M\) in the hyperbolic space \(H^{m + 1}\) of constant curvature \(-1\) and denote the \(r\)th mean curvature of \(M\) by \(H_r\). The authors prove: a) If \(C\) is a sphere and \(H_r\) is a constant \(\in [0,1]\) for some \(r\), then \(M\) is part of a equidistant sphere of \(H^{m+1}\). b) If \(H_1\) is constant, \(C\) is a convex submanifold of a geodesic hyperplane \(N \subset H^{m + 1}\), and if \(M\) is transverse to \(N\) along \(C\), then \(M\) has all the symmetries of \(C\). For the last result a flux formula for Killing vector fields of the hyperbolic space is derived. Reviewer: H.Reckziegel (Köln) Cited in 5 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:symmetries; constant mean curvature hypersurfaces; flux formula; hyperbolic space PDF BibTeX XML Cite \textit{B. Nelli} and \textit{H. Rosenberg}, Ann. Global Anal. Geom. 13, No. 1, 23--30 (1995; Zbl 0831.53039) Full Text: DOI References: [1] Barbosa, J.L.: Constant Mean Curvature Surfaces Bounded by a Planar Curve.Matematica Contemporanea 1 (1991), 3-15. · Zbl 0854.53010 [2] Barbosa, J.L.; Jorge, L.P.: StableH-Surfaces SpanningS 1 (1). To appear in:An. Acad. Bras. Ciênc. (1994). [3] Brito, F.;Meeks III, W.H.;Rosenberg, H.;Sa Earp, R.: Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve.Indiana Univ. Math. J. 40 (1991) 1, 333-343. · Zbl 0732.35068 · doi:10.1512/iumj.1991.40.40001 [4] Hopf, H.:Differential Geometry in the Large. Lecture Notes in Mathematics 1000, Springer-Verlag, 1983. · Zbl 0526.53002 [5] de Miranda Gomes, J.:Sobre hipersuperficies com curvatura media constante no espaco hiperbolico. PhD thesis IMPA, 1985. [6] Kapouleas, N.: Compact Constant Mean Curvature Surfaces in Euclidean Three-Space.J. Differ. Geom. 33 (1991), 683-715. · Zbl 0727.53063 [7] Nelli, B.; Sa Earp, R.: Some Properties of Hypersurfaces of Prescribed Mean Curvature in ? n+1. To appear in:Bull. Sci. Math., II. Sér. · Zbl 0872.53008 [8] Rosenberg, H.: Hypersurfaces of Constant Curvature in Space Forms.Bull. Sci. Math., II. Sér. 117 (1993), 211-239. · Zbl 0787.53046 [9] Rosenberg, H.; Spruck, J.: On the Existence of Convex Hypersurfaces of Constant Gauss Curvature in Hyperbolic Space. To appear in:J. Differ. Geom. · Zbl 0823.53047 [10] Spivak, M.:A Comprehensive Introduction to Differential Geometry IV. Publish or Perish Inc., Berkley 1979. · Zbl 0439.53004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.