[Paper Review] Some open questions on anti-de Sitter geometry

Thierry Barbot, Francesco Bonsante+7 more|arXiv (Cornell University)|May 28, 2012

Geometric and Algebraic Topology41 references27 citations

TL;DR

This paper presents a comprehensive list of open problems in anti-de Sitter (AdS) geometry, focusing on locally homogeneous Lorentzian manifolds of constant curvature −1. It explores connections to Teichmüller theory, discrete group actions, hyperbolic geometry analogs, and physical models such as multi-black holes and massive particles, with key contributions in convex core geometry, proper actions on complex AdS space, and maximal surfaces in 3D AdS manifolds.

ABSTRACT

We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\\"uller theory, as well as analogs in hyperbolic geometry.

Motivation & Objective

• To compile and systematize a wide range of open questions in anti-de Sitter (AdS) geometry, particularly in dimension 3 and higher.
• To highlight deep connections between AdS geometry and other mathematical areas, including Teichmüller theory, discrete group actions, and hyperbolic geometry.
• To explore physical motivations such as globally hyperbolic spacetimes, multi-black holes, and massive particles (tachyons) in AdS.
• To investigate the role of complex AdS space $X_{\mathbb{C}}$ as a complexification of both $\mathrm{AdS}_3$ and $\mathbb{H}^3$, and its relevance to holomorphic Riemannian geometry.
• To identify geometric and dynamical conditions under which discrete groups act properly discontinuously on $X_{\mathbb{C}}$, especially via pairs of quasi-Fuchsian or convex cocompact representations.

Proposed method

• Use of the standard model of $\mathrm{AdS}_n$ as the quadric $\{x \in \mathbb{R}^{n-1,2} \mid b(x,x) = -1\}$ with the Lorentzian metric induced by the bilinear form of signature $(n-1,2)$.
• Leveraging the isomorphism $\mathrm{AdS}_n \cong \mathrm{O}(n-1,2)/\mathrm{O}(n-1,1)$ to study locally homogeneous structures and Clifford–Klein forms.
• Applying techniques from Teichmüller theory and measured lamination spaces to study convex cores and boundary metrics in 3D AdS manifolds.
• Analyzing geometric time functions and $F$-time functions to understand foliations and causal structure in AdS spacetimes.
• Using the complexification $X_{\mathbb{C}} = \{z \in \mathbb{C}^4 \mid b(z,z) = -1\}$ as a model space for holomorphic Riemannian 3-manifolds of constant curvature.
• Investigating proper discontinuous actions of $\rho_l \times \rho_r(\pi_1(S)) \subset \mathrm{PSL}_2(\mathbb{C}) \times \mathrm{PSL}_2(\mathbb{C})$ on $X_{\mathbb{C}}$ via length and Lipschitz invariants $C_{\text{length}}$ and $C_{\text{Lip}}$.

Experimental results

Research questions

• RQ1 Can every pair of quasi-Fuchsian representations of a surface group be realized via an equivariant space-like embedding of the universal cover into complex AdS space $X_{\mathbb{C}}$?
• RQ2 For a pair of convex cocompact representations $\rho_l, \rho_r$ of a hyperbolic 3-manifold's fundamental group, does there exist a maximal domain in $X_{\mathbb{C}}$ where the action of $\rho_l \times \rho_r(\pi_1(M))$ is properly discontinuous?
• RQ3 What geometric conditions determine whether the action of $\rho_l \times \rho_r(\pi_1(M))$ on $X_{\mathbb{C}}$ is proper, and how can this be characterized via $C_{\text{length}}(\rho_l, \rho_r) < 1$?
• RQ4 Can the induced metric or measured bending lamination on the boundary of the convex core of a 3D AdS manifold be prescribed arbitrarily?
• RQ5 What is the relationship between the volume and width of the convex core in globally hyperbolic AdS 3-manifolds, and are there minima under deformation?

Key findings

• The complex AdS space $X_{\mathbb{C}}$ serves as a common complexification of both $\mathrm{AdS}_3$ and $\mathbb{H}^3$, with isometry group $\mathrm{O}(4,\mathbb{C}) \cong \mathrm{PSL}_2(\mathbb{C}) \times \mathrm{PSL}_2(\mathbb{C})$ up to finite index.
• For convex cocompact representations $\rho_l, \rho_r$, the action of $\rho_l \times \rho_r(\pi_1(M))$ on $X_{\mathbb{C}}$ is properly discontinuous if and only if $C_{\text{length}}(\rho_l, \rho_r) < 1$.
• There exist pairs of convex cocompact representations for which the action on $X_{\mathbb{C}}$ is properly discontinuous on a maximal domain, even though it is not proper on the entire space.
• The existence of a tachyon singularity along a closed curve in the unit tangent bundle of a hyperbolic surface cannot be achieved via small deformations of the standard AdS structure.
• The volume of the convex core in 3D AdS manifolds is conjectured to be minimized under certain geometric constraints, and its convexity under deformation remains an open question.
• Maximal surfaces in AdS 3-manifolds give rise to symplectic maps and harmonic extensions of quasi-symmetric homeomorphisms of the circle, linking to Teichmüller theory.