Abstract
A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ⊆ Hn+1 be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further Hr ≡ c for one normalized mean curvature. Then Mn is a horosphere or a geodesic cylinder if\(r{\leq}\ {2\over 3}\ (n+1)\). For \(r >\ {2\over 3}\ (n+1)\) the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions.
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Kohlmann, P. Uniqueness of horospheres and geodesic cylinders in hyperbolic space. Results. Math. 36, 75–101 (1999). https://doi.org/10.1007/BF03322104
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DOI: https://doi.org/10.1007/BF03322104