Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

TL;DR

This paper characterizes quasiperiodic maximal surfaces in pseudo-hyperbolic spaces through curvature and hyperbolicity conditions, linking their limit curves to quasisymmetric parametrizations and exploring applications in hyperbolic geometry and Teichmüller theory.

Contribution

It introduces a new characterization of quasiperiodic maximal surfaces via curvature and hyperbolicity, and establishes a correspondence between limit curves and quasisymmetric parametrizations.

Findings

Limit curves admit canonical quasisymmetric parametrizations.

Every quasisymmetric curve bounds a quasiperiodic surface with a continuous uniformization.

Applications to hyperbolic surfaces, rigidity of representations, and Teichmüller space.

Abstract

We study in this paper quasiperiodic maximal surfaces in pseudo-hyperbolic 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 Teichm\"uller space.