Convex surfaces with prescribed induced metrics in anti-de Sitter spacetimes

TL;DR

This paper proves the existence and uniqueness of certain convex surfaces with prescribed metrics in anti-de Sitter spacetimes, linking geometric structures on surfaces to spacetime geometry.

Contribution

It establishes a unique correspondence between prescribed surface metrics and quasifuchsian AdS spacetimes with convex Cauchy surfaces.

Findings

Existence of a unique quasifuchsian AdS spacetime for given metrics

Construction of past-convex Cauchy surfaces with prescribed induced metrics

Link between surface geometry and spacetime structure

Abstract

Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric of curvature $K<-1$ on $S$, and let $h_0$ be a hyperbolic metric on $S$. We show that there exists a unique quasifuchsian AdS spacetime with left metric isotopic to $h_0$, containing a past-convex Cauchy surface with induced metric isotopic to $h$.