Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

Abstract

We study in this paper quasiperiodic maximal surfaces in pseudohyperbolic spaces and show that they are characterised by a curvature condition, Gromov hyperbolicity or conformal hyperbolicity. We show that the limit curves of these surfaces in the Einstein Universe admits a canonical quasisymmetric parametrisation, while conversely every quasisymmetric curve in the Einstein Universe bounds a quasiperiodic surface in such a way that the quasisymmetric parametrisation is a continuous extension of the uniformisation; we give applications of these results to asymptotically hyperbolic surfaces, rigidity of Anosov representations and a version of the universal Teichmüller space.

Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces

“…This result is closely related to a recent work of Labourie and Toulisse [19] on the maximal surfaces in H 2,2 . We also generalize the existence and uniqueness of bounded solutions to general cyclic Higgs bundles in a subsequent paper [23,Section 7].…”

Simple length rigidity for Hitchin representations

“…Finally, we recover Anosov representations as periodic cases of uniformly hyperbolic bundles. Uniformly hyperbolic bundle is the structure underlying the study of quasi-symmetric maps in [12].…”

“…Consequently, for any section D of F 0 , we write D = D + + D − where D ± are sections of F ± 0 according to the decomposition (12).…”