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Existence and regularity of constant mean curvature hypersurfaces in hyperbolic space. (English) Zbl 0874.53050
Let \(\mathbb{H}^{n+1}\) denote the upper halfspace model of the hyperbolic space and let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\subset \mathbb{H}^{n+1}_\infty\) where \(\mathbb{H}^{n+1}_\infty\) is the sphere at infinity of \(\mathbb{H}^{n+1}\). Assume that \(\partial\Omega\) is \(C^1\). If \(H\in \mathbb{R}\) with \(|H|<1\), then there is a hypersurface \(M\) with small singular set having constant mean curvature \(H\) with respect to the inward pointing normal and boundary at infinity equal to \(\partial \Omega\). If \(\partial \Omega\) is in class \(C^{k,\alpha}\), \(1\leq k\leq n-1\) and \(0\leq \alpha\leq 1\) or \(k=n\) and \(0\leq \alpha< 1\), then \(M\cup\partial\Omega\) is close to infinity a \(C^{k,\alpha}\) submanifold with boundary. It is shown that \(\partial\Omega\) in \(C^{k,\alpha}\) for \(k\geq n+1\) and \(0<\alpha<1\) and \(H=0\) implies that \(M\cup \partial\Omega\) is close to infinity a \(C^{k,\alpha}\) submanifold with boundary if \(n\) is even, but not necessarily if \(n\) is odd. The corresponding study for minimal hypersurfaces is due to M. T. Anderson (existence) [Invent. Math. 69, 477-494 (1982; Zbl 0515.53042); Comment. Math. Helv. 58, 264-290 (1983; Zbl 0549.53058)] and R. Hardt and F.-H. Lin (regularity), see, e.g., F. H. Lin [Commun. Pure Appl. Math. 42, 229-242 (1989; Zbl 0688.49042)].

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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