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Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes

Roman Prosanov

TL;DR

This work solves polyhedral Alexandrov–Weyl-type questions for GHMC anti-de Sitter (2+1)-spacetimes by realizing concave hyperbolic cone-metrics on a genus g ≥ 2 surface as future-convex bent Cauchy surfaces inside a spacetime with prescribed left holonomy, and by realizing two cone-metrics simultaneously as a future- and past-convex pair of bent surfaces. The key technique is a continuity method implemented via blow-ups of deformation spaces and a geometric transition between AdS and Minkowski geometries, leveraging a sharp intrinsic-metric map that ties surface geometry to cone-metric data. Central to the construction are bent surfaces (a relaxed, polyhedral notion) and a K-surface foliation, enabling control of degenerations and enabling properness arguments that yield existence and local uniqueness in a strong neighborhood of zero cone-metrics. The results advance the understanding of rigidity for convex-like slices in AdS spacetimes, connect Teichmüller-theoretic data to spacetime geometry through earthquake theory, and showcase the efficacy of projective-geometric methods (transitions, blow-ups) in Lorentzian 3-manifold geometry with potential implications for quantum gravity models and holographic correspondences in low dimensions.

Abstract

We first prove that given a Fuchsian representation $ρ_\circ: π_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2π$ isometrically embeds as a future-convex bent Cauchy surface in a globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter (2+1)-spacetime whose left representation is $ρ_\circ$. Second, we show that given any two such cone-metrics, there exists a GHMC anti-de Sitter (2+1)-spacetime in which the cone-metrics embed simultaneously, one as a future-convex bent Cauchy surface and one as a past-convex. Furthermore, in both cases we establish that such a spacetime and embeddings are unique provided that the cone-metrics are sufficiently small.

Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes

TL;DR

This work solves polyhedral Alexandrov–Weyl-type questions for GHMC anti-de Sitter (2+1)-spacetimes by realizing concave hyperbolic cone-metrics on a genus g ≥ 2 surface as future-convex bent Cauchy surfaces inside a spacetime with prescribed left holonomy, and by realizing two cone-metrics simultaneously as a future- and past-convex pair of bent surfaces. The key technique is a continuity method implemented via blow-ups of deformation spaces and a geometric transition between AdS and Minkowski geometries, leveraging a sharp intrinsic-metric map that ties surface geometry to cone-metric data. Central to the construction are bent surfaces (a relaxed, polyhedral notion) and a K-surface foliation, enabling control of degenerations and enabling properness arguments that yield existence and local uniqueness in a strong neighborhood of zero cone-metrics. The results advance the understanding of rigidity for convex-like slices in AdS spacetimes, connect Teichmüller-theoretic data to spacetime geometry through earthquake theory, and showcase the efficacy of projective-geometric methods (transitions, blow-ups) in Lorentzian 3-manifold geometry with potential implications for quantum gravity models and holographic correspondences in low dimensions.

Abstract

We first prove that given a Fuchsian representation , where is a closed oriented surface of genus , any hyperbolic cone-metric on with cone-angles isometrically embeds as a future-convex bent Cauchy surface in a globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter (2+1)-spacetime whose left representation is . Second, we show that given any two such cone-metrics, there exists a GHMC anti-de Sitter (2+1)-spacetime in which the cone-metrics embed simultaneously, one as a future-convex bent Cauchy surface and one as a past-convex. Furthermore, in both cases we establish that such a spacetime and embeddings are unique provided that the cone-metrics are sufficiently small.

Paper Structure

This paper contains 35 sections, 107 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the results
  4. Proof ideas
  5. Context
  6. Notation
  7. Construction of blow-ups
  8. Cones and blow-ups
  9. The spaces of cone-metrics
  10. Blow-up of the space of cone-metrics
  11. Projective geometry of spacetimes of constant curvature
  12. Projective geometries and geometric transition
  13. Domains of discontinuity
  14. Convergence of domains of discontinuity
  15. Spaces of bent surfaces
  16. ...and 20 more sections

Key Result

Theorem AdS1

Let $S$ be a closed oriented surface of genus $\geq 2$, $V \subset S$ a finite non-empty set, $\rho_\circ: \pi_1 S \rightarrow G$ a Fuchsian representation and $d$ a concave hyperbolic cone-metric on $(S, V)$. Then there exist a GHMC anti-de Sitter (2+1)-spacetime $\Omega\cong S\times \mathbb{R}$ wh

Theorems & Definitions (158)

  • Theorem AdS1
  • Theorem AdS2
  • Theorem M1
  • Lemma AdS1
  • Lemma AdS2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 148 more