Abstract.
We give an existence result for constant mean curvature graphs in hyperbolic space \({\Bbb H}^{n+1}\). Let \(\Omega\) be a compact domain of a horosphere in \({\Bbb H}^{n+1}\) whose boundary \(\partial\Omega\) is mean convex, that is, its mean curvature \(H_{\partial\Omega}\) (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that \(-H_{\partial\Omega}< H < 1 \), then there exists a graph over \(\Omega\) with constant mean curvature H and boundary \(\partial\Omega\). Umbilical examples, when \(\partial\Omega\) is a sphere, show that our hypothesis on H is the best possible.
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Received July 18, 1997 / Accepted April 24, 1998
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López, R., Montiel, S. Existence of constant mean curvature graphs in hyperbolic space. Calc Var 8, 177–190 (1999). https://doi.org/10.1007/s005260050122
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DOI: https://doi.org/10.1007/s005260050122