The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries

Abstract

Let $h^{+}$ and $h^{-}$ be two complete, conformal metrics on the disc $\mathbb{D}$. Assume moreover that the derivatives of the conformal factors of the metrics $h^{+}$ and $h^{-}$ are bounded at any order with respect to the hyperbolic metric, and that the metrics have curvatures in the interval $\left(-\frac{1}ε, -1 - ε\right)$, for some $ε> 0$. Let $f$ be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset $Ω$ (see Definition 4.1) of the three-dimensional anti-de Sitter space, such that $Ω$ has $h^{+}$ (respectively $h^{-}$) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map $Φ_Ω$ (see Definition 5.7) equal to $f$.

Figures (3)

• Figure 1: In other words, Tamburelli has shown that for any closed surface $S$ with genus greater than or equal to $2$, and for any Riemannian metrics $h^{+}$ and $h^{-}$ on $S$ with curvatures strictly smaller than $-1$, there exists a maximal globally hyperbolic manifold $M$ homeomorphic to $S \times (0,1)$ and a submanifold $N$ of $M$ that has the same homotopy type as $S$ and is diffeomorphic to $S \times [0,1]$, such that the boundary of $N$ consists of two disjoint space-like surfaces on which $M$ induces a metric on $S \times \left\{ 0\right\}$ homotopic to $h^{+}$ and a metric on $S \times \left\{ 1\right\}$ homotopic to $h^{-}$.
• Figure 2: The connected components of $\partial \Omega \cap \mathbb{ADS}^{2,1}$, denoted by $\partial^{+} \Omega$ and $\partial^{-} \Omega$, are isometric to $(\mathbb{D}, h^{+})$ and $(\mathbb{D}, h^{-})$, respectively. The isometries $V^{+}$ and $V^{-}$ can be extended to the ideal boundary and then define the map $(\partial V^{-})^{-1} \circ \partial V^{+}$ (see Section \ref{['Gluing-maps-section']}), which is referred to as the gluing map of $\Omega$. In Theorem \ref{['mainn']} this gluing map is equal to $f$.
• Figure 3: If there is a closed hyperbolic surface $S_n$, two Fuchsian representations $\rho_{n}^{+}$ and $\rho_{n}^{-}: \pi_{1}(S_n) \to \rm{PSL}(2,\mathbb{R})$, and two complete Riemannian metrics $h^{+}_{n}$ and $h^{-}_{n}$ on $\mathbb{D}$ that have sectional curvature strictly less than $-1$ and are invariant under the action of $\rho_{n}^{+}$ and $\rho_{n}^{-}$ respectively. Then by Theorem \ref{['tumb']}, we can find a globally hyperbolic manifold diffeomorphic to $M_{n} = S_n \times [0,1]$ such that the induced metric on $S_n \times \left\{ 0\right\}$ is homotopic to $h^{+}_{n} / \rho_{n}^{+}$, and the induced metric on $S_n \times \left\{ 1\right\}$ is homotopic to $h^{-}_{n} / \rho_{n}^{-}$. The globally hyperbolic manifolds $M_{n}$ lift to globally hyperbolic convex subsets $\Omega_{n}$. There is a unique quasi-symmetric map equivariant under the action of $\rho^{+}_{n}$ and $\rho^{-}_{n}$, the gluing map of $\Omega_{n}$ must be equal to this map. The convex sets $\Omega_{n}$ will converge to the desired $\Omega$ that we want to realize.

Theorems & Definitions (9)

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