do Carmo, Manfredo Perdigão; Lawson, H. Blaine jun. On Alexandrov-Bernstein theorems in hyperbolic space. (English) Zbl 0534.53049 Duke Math. J. 50, 995-1003 (1983). The authors consider complete properly embedded hypersurfaces of constant mean curvature in hyperbolic n-space. Among them they characterize distance ”spheres” 1) around a point, 2) around a geodesic hypersurface and 3) horospheres by the behaviour of their asymptotic boundary. Reviewer: D.Ferus Cited in 4 ReviewsCited in 25 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A35 Non-Euclidean differential geometry Keywords:Berstein theorems; constant mean curvature; asymptotic boundary; horospheres PDFBibTeX XMLCite \textit{M. P. do Carmo} and \textit{H. B. Lawson jun.}, Duke Math. J. 50, 995--1003 (1983; Zbl 0534.53049) Full Text: DOI References: [1] Alexandrov, A characteristic property of spheres , Ann. Mat. Pura Appl. (4) 58 (1962), 303-315. · Zbl 0107.15603 · doi:10.1007/BF02413056 [2] F. J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem , Ann. of Math. (2) 84 (1966), 277-292. JSTOR: · Zbl 0146.11905 · doi:10.2307/1970520 [3] M. Anderson, Complete minimal varieties in hyperbolic space , (to appear). · Zbl 0515.53042 · doi:10.1007/BF01389365 [4] S. N. Bernstein, Sur un théorème de géometrie et son application aux équations aux dérivées partielles du type elliptique , Soobsc. Har’kov. Math. Obsc. 15 (1915), 38-45. · JFM 48.1401.01 [5] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem , Invent. Math. 7 (1969), 243-268. · Zbl 0183.25901 · doi:10.1007/BF01404309 [6] E. de Giorgi, Una estensione del teorema di Bernstein , Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 79-85. · Zbl 0168.09802 [7] J. de Miranda Gomes, Parabolic surfaces of constant mean curvature , I.M.P.A. preprint, 1983. [8] W. H. Fleming, On the oriented Plateau problem , Rend. Circ. Mat. Palermo (2) 11 (1962), 69-90. · Zbl 0107.31304 · doi:10.1007/BF02849427 [9] K. R. Frensel, O princípio da tangência e suas aplicações , Master’s thesis, IMPA, Rio de Janeiro, 1983. [10] Y. Hsiang, Z. H. Teng, and W. Yu, New examples of constant mean curvature immersions of \((2k-1)\)-spheres into euclidean \(2k\)-space , to appear in Ann. of Math. · Zbl 0522.53052 [11] H. B. Lawson, Jr., Complete minimal surfaces in \(S^3\) , Ann. of Math. (2) 92 (1970), 335-374. JSTOR: · Zbl 0205.52001 · doi:10.2307/1970625 [12] J. Simons, Minimal varieties in riemannian manifolds , Ann. of Math. (2) 88 (1968), 62-105. JSTOR: · Zbl 0181.49702 · doi:10.2307/1970556 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.