A Hilbert metric for bounded symmetric domains

Elisha Falbel (1, 2) , Antonin Guilloux (1, 3, 2) , Pierre Will (3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
3 IF - Institut Fourier

Bounded symmetric domains carry several natural invariant metrics, for example the Carath\'eodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish-Chandra realization in projective spaces. We describe this construction on the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to Carath\'eodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

Mots clés: Hilbert metric; Bounded symmetric domains

Domaines: Géométrie différentielle [math.DG]; Variables complexes [math.CV]; Géométrie métrique [math.MG]

Elisha Falbel, Antonin Guilloux, Pierre Will. A Hilbert metric for bounded symmetric domains. 2024. ⟨hal-04524876⟩