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arXiv · 2010.05704

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

François Labourie·Jérémy Toulisse
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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.

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François Labourie, Jérémy Toulisse. 2020-10-12. Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces. https://arxiv.org/abs/2010.05704

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