Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

TL;DR

This paper proves the existence of smooth isometric embeddings of certain conformal surfaces into hyperbolic and anti-de Sitter spaces, extending to boundary homeomorphisms with prescribed boundary curves.

Contribution

It establishes new existence results for embedded convex surfaces in hyperbolic and anti-de Sitter spaces with prescribed boundary at infinity.

Findings

Existence of smooth isometric embeddings for given boundary curves.

Extensions of embeddings to boundary homeomorphisms.

Applicability to Jordan curves in hyperbolic space.

Abstract

We show that given a quasi-circle $C$ in $\partial_{\infty}\mathbb{H}^3$ (respectively in $\partial_{\infty} \mathbb{ADS}^3$) 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 $V : (\mathbb{D}, h) \to \mathbb{ADS}^3$) such that $V$ extends continuously to a homeomorphism $\partial V : \partial_{\infty}\mathbb{H}^2 \to C$. In the case of hyperbolic space, the statement still holds if $C$ is a Jordan curve.