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

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.