# The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

## TL;DR

This paper explores the relationship between boundary metrics of convex regions in hyperbolic and anti-de Sitter spaces, showing that any quasisymmetric map can be realized as a boundary gluing map for certain geometric configurations.

## Contribution

It generalizes classical results by demonstrating that all quasisymmetric maps can be realized as boundary gluings in unbounded convex sets with specified curvature conditions.

## Key findings

- Any quasisymmetric map is achievable as a boundary gluing map for some quasicircle.
- Results extend classical boundary metric characterizations to unbounded convex regions.
- Analogous theorems are established in anti-de Sitter geometry.

## Abstract

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature $K \in [-1,0)$ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of $K$, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes.