On the Asymptotic Plateau Problem for CMC Hypersurfaces in Hyperbolic Space

Abstract

Let $$\mathbb {R}_{+}^{n+1}$$ be the half-space model of the hyperbolic space $$\mathbb {H}^{n+1}.$$ It is proved that if $$\Gamma \subset \left\{ x_{n+1} =0\right\} \subset \partial _{\infty }\mathbb {H}^{n+1}$$ is a bounded $$C^{0}$$ Euclidean graph over $$\left\{ x_{1}=0, x_{n+1}=0\right\} $$ then, given $$\left| H\right| <1,$$ there is a complete, properly embedded, CMC H hypersurface $$\Sigma $$ of $$\mathbb {H}^{n+1}$$ such that $$\partial _{\infty }S=\Gamma \cup \left\{ x_{n+1}=+\infty \right\} .$$ This result can be seen as a limit case of the existence theorem proved by Guan and Spruck in Guan and Spruck (2000) on CMC $$\left| H\right| <1$$ radial graphs with prescribed $$C^{0}$$ asymptotic boundary data. In spite of the above presentation of our result, our proof does not use coordinates but the Killing graph approach and therefore not only does not depend on the model of $$\mathbb {H}^{n},$$ but also allows the use of natural intrinsic geometric barriers of the hyperbolic space. A simple adaptation of our proof gives a new proof of Theorem 4.8 of Guan and Spruck (2000) and Theorem 4 of Dajczer et al. (2016). By the recent result on interior gradient estimates obtained by Marcos Dajzcer, Jorge Lira and the first author in Dajczer et al. (2016) we are able to apply here Perron’s method instead of the exhaustion method traditionally used in papers dealing with asymptotic Dirichlet problems, as in Guan and Spruck (2000) and Dajczer et al. (2016). The possibility of using Perron’s method is fundamental since the exhaustion technique seems strongly to not work for “horizontal” graphs.