On the geometry of horospheres
Many characterizations of horospheres have been found. So for instance:NEWLINENEWLINENEWLINE (a) \textit{M. P. do Carmo} and \textit{H. B. Lawson jun.} [Duke Math. J. 50, 995--1003 (1983; Zbl 0534.53049)] proved that any complete hypersurface properly embedded with constant mean curvature in \(\mathbb{H}^{n+1}\) with a single point at the asymptotic boundary is a horosphere. NEWLINENEWLINENEWLINE (b) \textit{L. J. Alías} and \textit{M. Dajczer} [Comment. Math. Helv. 81, No. 3, 653--663 (2006; Zbl 1110.53039)] proved that a surface properly immersed in \(\mathbb{H}^{3}\) with constant mean curvature \(-1\leq H \leq 1\) and contained in a slab must be a horosphere. NEWLINENEWLINENEWLINEThe aim of this paper is to find reasonable geometric conditions for complete hypersurfaces immersed in \(\mathbb{H}^{n+1}\) as a horosphere. By the method of generalized maximum principles, geometry of support function and Lorentz Gauss mapping, the authors prove that horospheres and hyperbolic cylinders are the only complete hypersurfaces with constant mean curvature of \(\mathbb{H}^{n+1}\) whose support functions determined by a nonzero null vector are linearly related. This result resolves the case that appeared in their earlier paper [J. Math. Anal. Appl. 386, No. 2, 862--869 (2012; Zbl 1251.53035)].
scientific article; zbMATH DE number 6361415
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