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

Labourie, François

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 439-471.

Nous étudions deux classes de représentations linéaires d'un groupe de surface  : les représentations de Hitchin et les représentations symplectiques maximales. En reliant ces représentations à des birapports, nous montrons qu'elles sont déplaçantes, c'est-à-dire que leurs longueurs de translation sont grossièrement contrôlées par celles du graphe de Cayley. Ceci nous permet de montrer que le groupe modulaire agit proprement sur l'espace de ces représentations et que la fonctionnelle énergie associée à une telle représentation est propre. Nous en déduisons alors l'existence de surfaces minimales dans les quotients d'espaces symétriques associés et en tirons deux conséquences  : un résultat de rigidité pour les représentations symplectiques et un résultat partiel concernant la description de la composante de Hitchin en termes purement holomorphes.

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.

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