Constant Gauss curvature foliations of AdS spacetimes with particles

TL;DR

This paper proves the existence and uniqueness of constant Gauss curvature foliations in convex GHM AdS spacetimes with particles, and relates the space of such metrics to Teichmüller spaces with cone singularities.

Contribution

It extends previous results by providing a new proof for the foliation existence and describes a parametrization of the space of convex GHM AdS metrics with particles.

Findings

Unique foliation by constant Gauss curvature surfaces in the complement of the convex core.

Parametrization of convex GHM AdS metrics via Teichmüller spaces with cone singularities.

Extension of landslides and earthquake analogs to hyperbolic surfaces with cone singularities.

Abstract

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $\pi$ along time-like curves), the complement of the convex core in $N$ admits a unique foliation by constant Gauss curvature surfaces. This extends, and provides a new proof of, a result of \cite{BBZ2}. We also describe a parametrization of the space of convex GHM AdS metrics on a given manifold, with particles of given angles, by the product of two copies of the Teichm\"uller space of hyperbolic metrics with cone singularities of fixed angles. Finally, we use the results on $K$-surfaces to extend to hyperbolic surfaces with cone singularities of angles less than $\pi$ a number of results concerning landslides, which are smoother analogs of earthquakes sharing some of their key properties.