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

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.

PDF Access Denied

We have not been able to recognize your IP address 3.133.119.66 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

quasicircles, induced metric, Weyl problem

Mathematical Subject Classification 2010

Primary: 30F40, 53B30, 53C45, 53C50, 57M50

Secondary: 30C62, 37F30

References

Publication

Received: 14 February 2019

Revised: 12 August 2020

Accepted: 12 October 2020

Published: 30 November 2021

Proposed: Ian Agol

Seconded: Tobias H Colding, Anna Wienhard

Authors

Francesco Bonsante
Dipartimento di Matematica Università degli Studi di Pavia Pavia Italy
https://www-dimat.unipv.it/~bonsante
Jeffrey Danciger
Department of Mathematics University of Texas at Austin Austin, TX United States
https://www.ma.utexas.edu/users/jdanciger
Sara Maloni
Department of Mathematics University of Virginia Charlottesville, VA United States
https://sites.google.com/view/sara-maloni/
Jean-Marc Schlenker
Department of Mathematics University of Luxembourg Esch-sur-Alzette Luxembourg
https://math.uni.lu/schlenker
Boubacar Diallo

Volume 25, issue 6 (2021)

Francesco Bonsante, Jeffrey Danciger, Sara Maloni and Jean-Marc Schlenker

Appendix: Boubacar Diallo

Abstract