Variations along the Fuchsian locus

Labourie, François; Wentworth, Richard

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 2, pp. 487-547.

Notre résultat principal est une expression explicite de la métrique de pression sur la composante de Hitchin de l'espace des représentations du groupe fondamental d'une surface dans $\mathrm{𝖯𝖲𝖫}(n,ℝ)$ le long du lieu fuchsien. Cette formule utilise une paramétrisation de l'espace tangent à la composante de Hitchin en terme de différentielles holomorphes, et elle s'exprime explicitement en fonction du produit de Petersson. Au passage, nous établissons des relations qui généralisent les résultats classiques de la théorie de Teichmüller, tels que la formule de Gardiner, le rapport entre fonctions de longueur et déformations de Fenchel-Nielsen et les variations des birapports.

The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations into $\mathrm{𝖯𝖲𝖫}(n,ℝ)$ along the Fuchsian locus. The expression is in terms of a parametrization of the tangent space by holomorphic differentials, and it gives a precise relationship with the Petersson pairing. Along the way, variational formulas are established that generalize results from classical Teichmüller theory, such as Gardiner's formula, the relationship between length functions and Fenchel-Nielsen deformations, and variations of cross ratios.

Classification : 37D35; 58D29, 32G15, 14D20.
Keywords: Pressure metric, higher Teichmüller space, Gardiner formula, Higgs bundles, Hitchin components.
Mot clés : Métrique de pression, espace de Teichmüller généralisé, formule de Gardiner, fibrés de Higgs, composantes de Hitchin.

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