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A Hilbert metric for bounded symmetric domains

Elisha Falbel, Antonin Guilloux, Pierre Will

Abstract

Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, 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 Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

A Hilbert metric for bounded symmetric domains

Abstract

Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, 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 Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.
Paper Structure (19 sections, 21 theorems, 77 equations, 2 figures)

This paper contains 19 sections, 21 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Bounded symmetric domains and the generalized Hilbert metric
  3. A special case: the bidisc
  4. Borel and Harish-Chandra embeddings
  5. The Shilov boundary as the set $\Lambda$
  6. Transitivity and computation of the Hilbert metric
  7. Classical bounded symmetric domains
  8. Bounded symmetric domains and Hermitian symmetric spaces
  9. Hermitian symmetric spaces
  10. Bounded symmetric domains
  11. Projective embeddings
  12. Borel and Harish-Chandra embeddings
  13. Types $\mathrm{I}$, $\mathrm{II}$, $\mathrm{III}$
  14. Type $\mathrm{IV}$
  15. Singular value decomposition and the geometry of classical Hermitian symmetric spaces
  16. ...and 4 more sections

Key Result

Theorem 1

For any classical bounded symmetric domain $D$, the semi-metric $d_D$ is an actual metric, invariant by the automorphism group $\mathrm{Aut}(D)$ and comes from a Finsler infinitesimal metric.

Figures (2)

  • Figure 1: A synthetic view of the Borel and Harish-Chandra embeddings and the maps $E$ and $F$ (in red) for Type $\mathrm{I}$. Here, we have $W_x^+ \in G(p,n)$, $W_x^- \in G(q,n)$, $Z_x \in D^\mathrm{I}_{p,q}\subset M_{p,q}$ and $Z_x^* \in M_{q,p}$, $F(x) = E_q(x)\in \mathbb {P}(\Lambda^q\mathbb C^n)$ and $E(x) = E_p(x)\in \mathbb {P}(\Lambda^p\mathbb C^n)$. The black arrows are the different steps of Borel, Harish-Chandra and Plücker embeddings, denoted by respectively the letters $B$, $HC$ and $P$. The three blue arrows are given by orthogonality: taking the adjoint on the bottom left, orthogonality for the Hermitian structure in the middle, and orthogonality in duality in the bottom right.
  • Figure 2: The two ellipses $\mathcal{E}_1$ and $\mathcal{E}_2$ and the min and max values for $|V_1-V_2|$.

Theorems & Definitions (33)

  • Theorem
  • Proposition 1.1
  • Definition 2.1
  • Theorem : Borel embedding theorem
  • Definition 2.2
  • Theorem 2.3: Harish-Chandra
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 23 more