Möbius Structures, Hyperbolic Ends and k-Surfaces in Hyperbolic Space

Möbius surfaces and hyperbolic ends are key tools used in the study of geometrically finite three-dimensional hyperbolic manifolds. We review the theory of Möbius surfaces and describe a new framework for the theory of hyperbolic ends. We construct the ideal boundary functor sending hyperbolic ends into Möbius surfaces, and the extension functor sending Möbius surfaces into hyperbolic ends. We show that the former is a right inverse of the latter, and we show that every hyperbolic end canonically embeds into the extension of its ideal boundary. We conclude by showing that, for any given Möbius surface, there exists a unique maximal hyperbolic end having that Möbius surface for its ideal boundary. We apply these theories to the study of infinitesimally strictly convex (ISC) surfaces in ℍ3 which are complete with respect to the sums of their first and third fundamental forms (called quasicomplete in the sequel). We prove a new a priori C0 estimate for such surfaces. We apply this estimate to provide a complete solution of a Plateau-type problem for surfaces of constant extrinsic curvature in ℍ3 posed by Labourie in 2000 (Invent Math 141:239–297). We conclude by describing new parametrisations of the spaces of quasicomplete, ISC, constant extrinsic curvature surfaces in ℍ3 by open subsets of spaces of holomorphic functions.