A half-space property for hypersurfaces in the hyperbolic space (Q6622148)

Hyperspheres in \(\mathbb{H}^{n+1}\) are totally umbilical hypersurfaces of \(\mathbb{H}^{n+1}\) isometric to the hyperbolic space \(\mathbb{H}^{n}\). They define a complete foliation of hyperbolic space. \textit{M. P. do Carmo} and \textit{H. B. Lawson jun.} [Duke Math. J. 50, 995--1003 (1983; Zbl 0534.53049)] have used Alexandrov's reflexion method to show that a complete hypersurface \(\Sigma^{n}\) in \(\mathbb{H}^{n+1}\), properly embedded with constant mean curvature, with asymptotic boundary being a sphere and such that \(\Sigma^{n}\) separates poles, must be a hypersphere.\N\NThis result inspired several other authors to obtain some rigidity results for hypersurfaces in hyperbolic space. For example, \textit{C. P. Aquino} and \textit{H. F. de Lima} [J. Math. Anal. Appl. 386, No. 2, 862--869 (2012; Zbl 1251.53035)] used the quadric model of hyperbolic space to prove that the hyperspheres are the only complete immersed hypersurfaces with constant mean curvature determined by a space-like vector \(a\) and whose Gauss image lies in a totally umbilical space-like hypersurface of de Sitter space determined by \(a\).\N\NThe paper under review extends this result as follows: \N\NTheorem. Let \(a\) be a unit space-like vector. The only complete noncompact \(1\)-minimal hypersurface immersed into hyperbolic space \(\mathbb{H}^{n+1}\), with nonnegative mean curvature \(H\) and such that its distance \(d\) to the equator \(\mathbb{H}^{n}\) converges to zero at infinity, is the equator determined by \(a\).