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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)].

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