Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

Abstract

We show that given a quasi-circle $C\subset {\partial }_{\infty }{\mathbb{H}}^3$ (respectively $C\subset {\partial }_{\infty }\mathbb{AD}{\mathbb{S}}^3$) and a complete conformal metric $h$ on $\mathbb{D}$ whose curvature $K_h$ takes values in a compact subset of $(-1,0)$ (respectively $(-\infty ,-1)$), with all derivatives bounded with respect to the hyperbolic metric, there exists a smooth isometric embedding $V:(\mathbb{D},h)\to {\mathbb{H}}^3$ (respectively $V:(\mathbb{D},h)\to \mathbb{AD}{\mathbb{S}}^3$) such that $V$ extends continuously to a homeomorphism $\partial V:{\mathbb{S}}^1\to C$. In the hyperbolic case, the conclusion still holds if $C$ is an arbitrary Jordan curve.

14 pages, 1 figure. Corrections and Modifications Made in Response to the Referee's Report, to Appear in Proceedings of the American Mathematical Society