On (2,1)-dimensional anti-de Sitter spacetimes with convex boundary
Abstract
Let S be a closed oriented surface of genus at least $2$. Let X be a (2,1)-dimensional anti-de Sitter spacetime homeomorphic to S×[−1,1] with smooth, spacelike, strongly convex boundary. By Gauss' formula, the induced metric on each of the two components of ∂ X has curvature K<−1. In this paper we address the inverse problem. We show that, for any two smooth metrics h-, h+ on S with curvature K<−1, there exists a unique, smooth anti-de Sitter metric on X:=S× [−1,1] with respect to which the boundary is locally strongly convex and has induced metric isotopic to h- on its past component and isotopic to h+ on its future component. Existence has already been proven by Tamburelli. We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations l-, l+∈ℳℒS that together fill S, there exists a unique (2,1)-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively l- and l+. Here, existence has already been proven by Bonsante--Schlenker.
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BibTeX
@misc{prosanov202621dimensional,
title = {{On $(2,1)$-dimensional anti-de Sitter spacetimes with convex boundary}},
author = {Roman Prosanov and Jean-Marc Schlenker and Graham Andrew Smith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05506},
eprint = {2610.05506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05506},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On (2,1)-dimensional anti-de Sitter spacetimes with convex boundary
AU - Roman Prosanov
AU - Jean-Marc Schlenker
AU - Graham Andrew Smith
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05506
UR - https://arxiv.org/abs/2610.05506
AB - Let S be a closed oriented surface of genus at least $2$. Let X be a (2,1)-dimensional anti-de Sitter spacetime homeomorphic to S×[−1,1] with smooth, spacelike, strongly convex boundary. By Gauss' formula, the induced metric on each of the two components of ∂ X has curvature K<−1. In this paper we address the inverse problem. We show that, for any two smooth metrics h_-, h_+ on S with curvature K<−1, there exists a unique, smooth anti-de Sitter metric on X:=S× [−1,1] with respect to which the boundary is locally strongly convex and has induced metric isotopic to h_- on its past component and isotopic to h_+ on its future component. Existence has already been proven by Tamburelli. We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations l_-, l_+∈ℳℒ_S that together fill S, there exists a unique (2,1)-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively l_- and l_+. Here, existence has already been proven by Bonsante--Schlenker.
ER -