Embedded convex surfaces in hyperbolic and anti-de Sitter spaces

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.

Figures (1)

• Figure 1: For us, the convex set $\mathcal{C}$ will always be a convex subset of $\overline{\mathbb{H}}^3$ whose boundary in $\mathbb{H}^3$ is a smooth surface $\widetilde{S}$, with ideal boundary $\partial_{\infty}\widetilde{S}$ equal to a quasi-circle $C$. Moreover, $\partial_{\infty}\mathcal{C}$ is given by $C \cup \Omega^{-}$, where $\Omega^{-}$ is the component of $\partial_{\infty}\mathbb{H}^3 \setminus C$ lying on the convex side of $\widetilde{S}$.

Theorems & Definitions (11)

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