Prescribing metrics on the boundary of AdS 3-manifolds

TL;DR

This paper proves the existence of Anti-de Sitter 3-manifolds with prescribed boundary metrics of curvature less than -1, establishing a boundary metric prescription problem in the context of AdS geometry.

Contribution

It demonstrates the existence of AdS manifolds with specified boundary metrics of curvature less than -1, utilizing duality between convex surfaces in AdS3.

Findings

Existence of AdS manifolds with prescribed boundary metrics.

Construction of convex boundary surfaces with given metrics.

Application of duality in AdS geometry.

Abstract

We prove that given two metrics $g_{+}$ and $g_{-}$ with curvature $\kappa <-1$ on a closed, oriented surface $S$ of genus $\tau\geq 2$, there exists an $AdS$ manifold $N$ with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of $\partial N$ are equal to $g_{+}$ and $g_{-}$. Using the duality between convex space-like surfaces in $AdS_{3}$, we obtain an equivalent result about the prescription of the third fundamental form.