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Minimizing the volume of globally hyperbolic anti-de Sitter 3-manifolds

Gabriele Mondello, Nicolas Tholozan

Abstract

In this paper we show that the volume of a maximal globally hyperbolic Cauchy-compact anti-de Sitter $3$-manifold $M$ is at least $π^2|χ(M)|$, and that this minimum value is attained if and only if $M$ is Fuchsian.

Minimizing the volume of globally hyperbolic anti-de Sitter 3-manifolds

Abstract

In this paper we show that the volume of a maximal globally hyperbolic Cauchy-compact anti-de Sitter -manifold is at least , and that this minimum value is attained if and only if is Fuchsian.
Paper Structure (6 sections, 7 theorems, 54 equations)

This paper contains 6 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. MGHC $\mathbb{A}\mathrm{d}\mathbb{S}$ $3$-manifolds and their CMC foliation
  4. Constant mean curvature foliations
  5. The Fuchsian case
  6. Comparison with the Fuchsian case

Key Result

Theorem A

Let $M$ be a MGHC $\mathbb{A}\mathrm{d}\mathbb{S}$$3$-manifold with Cauchy surface $\Sigma$ of negative Euler characteristic $\chi(\Sigma)$. Then with equality if and only if $M$ is Fuchsian.

Theorems & Definitions (16)

  • Theorem A
  • Conjecture B
  • Definition 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['gaussian']}
  • Theorem 2.3: Barbot--Béguin--Zeghib BBZ
  • Proposition 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • ...and 6 more