A Hilbert metric for bounded symmetric domains
-
Elisha Falbel
Abstract
Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or Bergman metric. We define another natural metric, from the generalized Hilbert metric defined in [4], 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 for the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to the Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.
Funding statement: The three authors are partially funded by ANR-23-CE40-0012.
-
Communicated by: R. Löwen
References
[1] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften. Springer 1999. MR1744486 Zbl 0988.5300110.1007/978-3-662-12494-9Search in Google Scholar
[2] B. Chabat, Introduction à l’analyse complexe. Tome 2. Mir, Moscow 1990. MR1089193 Zbl 0732.32001Search in Google Scholar
[3] J.-L. Clerc, Geometry of the Shilov boundary of a bounded symmetric domain. In: Geometry, integrability and quantization, vol. 10 of Geom. Integrability Quantization, 11–55, Avangard Prima, Sofia 2009. Reprint of J. Geom. Symmetry Phys. 13 (2009), 25–74. MR2757825 Zbl 1182.53046Search in Google Scholar
[4] E. Falbel, A. Guilloux, P. Will, Hilbert geometry without convexity. J. Geom. Anal. 30 (2020), 2865–2896. MR4105139 Zbl 1441.5200910.1007/s12220-020-00426-xSearch in Google Scholar
[5] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, volume 34 of Graduate Studies in Mathematics. Amer. Math. Soc. 2001. MR1834454 Zbl 0993.5300210.1090/gsm/034Search in Google Scholar
[6] M. Jarnicki, P. Pflug, Invariant distances and metrics in complex analysis, volume 9 of De Gruyter Expositions in Mathematics. De Gruyter 2013. MR3114789 Zbl 1273.3200210.1515/9783110253863Search in Google Scholar
[7] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, volume 2 of Pure and Applied Mathematics. Dekker 1970. MR277770 Zbl 0207.37902Search in Google Scholar
[8] N. Mok, Metric rigidity theorems on Hermitian locally symmetric manifolds, volume 6 of Series in Pure Mathematics. World Scientific Publishing Co., Teaneck, NJ 1989. MR1081948 Zbl 0912.3202610.1142/0773Search in Google Scholar
[9] K. Morita, On the kernel functions for symmetric domains. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 5 (1956), 190–212. MR88567 Zbl 0072.30001Search in Google Scholar
[10] I. Satake, Algebraic structures of symmetric domains, volume 4 of Kanô Memorial Lectures. Princeton Univ. Press 1980. MR591460 Zbl 0483.32017Search in Google Scholar
[11] M. Suzuki, The intrinsic metrics on the circular domains in ℂn Pacific J. Math. 112 (1984), 249–256. MR739149 Zbl 0535.3201410.2140/pjm.1984.112.249Search in Google Scholar
[12] A. K. Wienhard, Bounded cohomology and geometry, volume 368 of Bonner Mathematische Schriften. Mathematisches Institut, Universität Bonn, 2004. MR2205508 Zbl 1084.32013Search in Google Scholar
[13] J. A. Wolf, Fine structure of Hermitian symmetric spaces. In: Symmetric spaces (Short Courses, Washington Univ., St. Louis, 1969–1970), 271–357, Dekker 1972. MR404716 Zbl 0257.32014Search in Google Scholar
[14] A. M. Zimmer, Characterizing domains by the limit set of their automorphism group. Adv. Math. 308 (2017), 438–482. MR3600063 Zbl 1362.3201510.1016/j.aim.2016.12.017Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
Articles in the same Issue
- Frontmatter
- Combinatorics of stratified hyperbolic slices
- Godbersen’s conjecture for locally anti-blocking bodies
- A Hilbert metric for bounded symmetric domains
- On the generalized Suzuki curve
- The partition of PG(2, q3) arising from an order 3 planar collineation
- Well-rounded lattices from odd prime degree number fields in the ramified case
- Split Cayley hexagons via subalgebras of octonion algebras
- Relative Lipschitz saturation of complex algebraic varieties
- The prime grid contains arbitrarily large empty polygons
- The geometry of locally bounded rational functions
