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Abstract

Let $H^{n+1}$ be the $n+1$ dimensional hyperbolic space and $\partial H^{n+1}$ its boundary at infinity, which may be identified with $S^{n}$ in the ball model or $R^{n}$ in the half space model of hyperbolic space. By a  boundary value problem at infinity (or an asymptotic Plateau problem) we understand a solution to the following problem:

Let $\Omega \subset H^{n+1}$ be a smooth bounded domain, $\Gamma = \partial \Omega$ and a smooth symmetric function $f$ of $n$ variables be given. We seek a complete hypersurface $\Sigma$ of “constant curvature” in $H^{n+1}$ satisfying  (0.1) $f(\kappa [\Sigma]) = \sigma$, (0.2) $\partial \Sigma = \Gamma$  where $\kappa [\Sigma] = (\kappa _{1}, \cdots, \kappa _{n})$ denotes the hyperbolic principal curvatures of $\Sigma$ and $\sigma \in  (0,1)$  is a constant. Classical choices for the curvature function $f(\kappa [\Sigma])$ are the mean curvature $H$ and the Gauss curvature $K$.

Such problems, besides being intrinsically interesting, have a deep connection to the study of hyperbolic manifolds, through the pioneering work of Labourie. They are also related to the existence of constant mean curvature and constant Gauss curvature  foliations in Lorentz manifolds, which are of interest in General relativity.

In this talk we will survey our work on these problems for general curvature functions paying attention to the existence of foliations as the parameter $\sigma$ varies. Time permitting, we will also discuss a new mean curvature flow for starshaped boundaries at infinity.