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On the existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space. (English) Zbl 0936.35069
Jost, Jürgen (ed.), Geometric analysis and the calculus of variations. Dedicated to Stefan Hildebrandt on the occasion of his 60th birthday. Cambridge, MA: International Press. 253-266 (1996).
From the introduction: Let $$\Gamma$$ be an embedded codimension one submanifold of $$\partial_\infty \mathbb{H}^{n+1}$$ (the boundary at infinity of hyperbolic space). We prove that any such mean convex $$\Gamma$$ is the asymptotic boundary of a complete embedded hypersurface of $$\mathbb{H}^{n+1}$$ of constant mean curvature $$H$$, for each $$H\in(0,1)$$. We construct the desired $$M$$ as a limit of constant mean curvature graphs over a fixed compact domain in a horosphere, for constant boundary data. Thus an important part of our study concerns the existence and uniqueness of constant mean curvature hypersurfaces which are graphs over a bounded domain in a horosphere, whose boundary is mean convex.
For the entire collection see [Zbl 0914.00109].

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature