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

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

Appendix: Boubacar Diallo

Geometry & Topology 25 (2021) 2827–2911

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 [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.

quasicircles, induced metric, Weyl problem
Mathematical Subject Classification 2010
Primary: 30F40, 53B30, 53C45, 53C50, 57M50
Secondary: 30C62, 37F30
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
Francesco Bonsante
Dipartimento di Matematica
Università degli Studi di Pavia
Jeffrey Danciger
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Sara Maloni
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Jean-Marc Schlenker
Department of Mathematics
University of Luxembourg
Boubacar Diallo