Symmetry of constant mean curvature hypersurfaces in hyperbolic space (Q1069470)

The authors extend work of \textit{M. P. Do Carmo} and \textit{H. B. Lawson jun. } [ibid. 50, 995-1003 (1983; Zbl 0534.53049)] on such hypersurfaces. Proofs depend on R. Schoen's formulation of the maximum principle. Theorem: Let M be a constant mean curvature hypersurface such that its points at infinity, \(\partial_{\infty}M\), are contained in the sphere of points at infinity, \(\partial_{\infty}P\), of a totally geodesic hyperplane P. Then either M is trivial, i.e. a parallel hypersurface to P, or, M has P as a hyperplane of symmetry and meets each normal to P at most once on each side of P. - Similarly, for minimal M information is derived from assumptions on \(\partial_{\infty}M\).