On the existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space. (English) Zbl 0936.35069

Nelli, Barbara; Spruck, Joel

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

Keywords:

asymptotic boundary; constant mean curvature graphs; horosphere