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Proceedings of the American Mathematical Society

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Asymptotically Poincaré surfaces in quasi-Fuchsian manifolds

Author: Keaton Quinn
Journal: Proc. Amer. Math. Soc. 148 (2020), 1239-1253
MSC (2010): Primary 30F60; Secondary 53C42
Published electronically: November 19, 2019
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Abstract: We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective structure at infinity. As an application, we establish a conjecture of Labourie from [J. London Math. Soc. 45 (1992), pp. 549-565] regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.

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Keaton Quinn
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607

Received by editor(s): December 18, 2018
Received by editor(s) in revised form: July 31, 2019
Published electronically: November 19, 2019
Additional Notes: The author was partially supported in summer 2018 by a research assistantship under NSF DMS-1246844, RTG: Algebraic and Arithmetic Geometry, at the University of Illinois at Chicago.
Communicated by: Ken Bromberg
Article copyright: © Copyright 2019 Keaton Quinn