Cross ratios, Anosov representations and the energy functional on Teichmüller space

François Labourie

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 3, page 439-471
  • ISSN: 0012-9593

Abstract

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We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.

How to cite

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Labourie, François. "Cross ratios, Anosov representations and the energy functional on Teichmüller space." Annales scientifiques de l'École Normale Supérieure 41.3 (2008): 439-471. <http://eudml.org/doc/272110>.

@article{Labourie2008,
abstract = {We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.},
author = {Labourie, François},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Hitchin components; energy; cross ratio; Toledo invariant; harmonic mappings; minimal surfaces},
language = {eng},
number = {3},
pages = {439-471},
publisher = {Société mathématique de France},
title = {Cross ratios, Anosov representations and the energy functional on Teichmüller space},
url = {http://eudml.org/doc/272110},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Labourie, François
TI - Cross ratios, Anosov representations and the energy functional on Teichmüller space
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 3
SP - 439
EP - 471
AB - We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.
LA - eng
KW - Hitchin components; energy; cross ratio; Toledo invariant; harmonic mappings; minimal surfaces
UR - http://eudml.org/doc/272110
ER -

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