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February 2014 Interior curvature estimates and the asymptotic plateau problem in hyperbolic space
Bo Guan, Joel Spruck, Ling Xiao
J. Differential Geom. 96(2): 201-222 (February 2014). DOI: 10.4310/jdg/1393424917

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.”

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Bo Guan. Joel Spruck. Ling Xiao. "Interior curvature estimates and the asymptotic plateau problem in hyperbolic space." J. Differential Geom. 96 (2) 201 - 222, February 2014. https://doi.org/10.4310/jdg/1393424917

Information

Published: February 2014
First available in Project Euclid: 26 February 2014

zbMATH: 1287.53051
MathSciNet: MR3178439
Digital Object Identifier: 10.4310/jdg/1393424917

Rights: Copyright © 2014 Lehigh University

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Vol.96 • No. 2 • February 2014
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