Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus

TL;DR

This work proves that globally hyperbolic compact anti-de Sitter $(2+1)$-spacetimes with strictly convex boundary (smooth or polyhedral) are locally determined by boundary data, in a neighborhood of the Fuchsian holonomy. The authors reduce the infinitesimal rigidity problem to the Minkowski setting via the infinitesimal Pogorelov map and leverage established rigidity results (Smith; Fillastre–Prosanov) along with a Nash–Moser framework to obtain a local homeomorphism for the induced-boundary metric map. They formulate smooth and polyhedral versions, with corresponding spaces of embeddings and boundary metrics, and show that the boundary metric data yields a local parametrization of the moduli space near the Fuchsian locus. The results bridge AdS geometry with Minkowski and hyperbolic analogues, providing a near-Fuchsian rigidity theory and paving the way for further global realization results in both smooth and polyhedral regimes. The findings have implications for boundary-driven realizability of AdS spacetimes and contribute to the broader program of understanding convex boundary rigidity in Lorentzian constant-curvature geometries.

Abstract

We prove that globally hyperbolic compact anti-de Sitter (2+1)-spacetimes with strictly convex spacelike boundary that is either smooth or polyhedral and whose holonomy is close to Fuchsian are determined by the induced metric on the boundary.

Theorems & Definitions (49)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1
• Theorem 3.2