Asymptotically Poincaré surfaces in quasi-Fuchsian manifolds

2018/11/21 by Quinn, Keaton
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.08828

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 1992 regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.

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