Realizing metrics of curvature $\leq -1$ on closed surfaces in Fuchsian anti-de Sitter manifolds

TL;DR

This paper proves that any metric with curvature less than or equal to -1 on a closed surface of genus greater than one can be realized as a space-like convex surface in a 3D anti-de Sitter space, extending geometric realization results.

Contribution

It establishes a new realization theorem for metrics of curvature ≤ -1 on closed surfaces within Lorentzian anti-de Sitter manifolds, using approximation methods and prior immersion results.

Findings

Any Alexandrov metric with curvature ≤ -1 on a closed surface is realizable in AdS space.

The proof employs approximation and existing immersion theorems.

Extends classical surface realization results to Lorentzian anti-de Sitter geometry.

Abstract

We prove that any metric with curvature $\leq -1$ (in the sense of A. D. Alexandrov) on a closed surface of genus $>1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$. The proof is done by approximation, using a result about isometric immersion of smooth metrics by Labourie--Schlenker.