## Journal of Differential Geometry

### Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

#### Abstract

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $\mathbb{H}^{n+1}$ satisfying $f(k) = \sigma \in (0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover, if $\Gamma$ is (Euclidean) star-shaped, the solution is unique and also (Euclidean) star-shaped, while if $\Gamma$ is mean convex, the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a “global interior curvature estimate.”

#### Article information

Source
J. Differential Geom., Volume 96, Number 2 (2014), 201-222.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.jdg/1393424917

Digital Object Identifier
doi:10.4310/jdg/1393424917

Mathematical Reviews number (MathSciNet)
MR3178439

Zentralblatt MATH identifier
1287.53051

#### Citation

Guan, Bo; Spruck, Joel; Xiao, Ling. Interior curvature estimates and the asymptotic plateau problem in hyperbolic space. J. Differential Geom. 96 (2014), no. 2, 201--222. doi:10.4310/jdg/1393424917. https://projecteuclid.org/euclid.jdg/1393424917