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On geometric properties of holomorphic isometries between bounded symmetric domains

Shan Tai Chan

TL;DR

This work analyzes holomorphic isometries from the complex unit ball into irreducible bounded symmetric domains under Bergman-type metrics, revealing a rigid geometric structure of their images. By constructing a bounded balanced convex domain $D_f$ and a surjective holomorphic submersion $\widetilde{\pi}_f:D_f\to D$, the authors show that $f$ behaves as a holomorphic section with a holomorphic splitting of tangent spaces, enabling a precise description of how affine-linear sections are mapped. They establish the existence of isometries whose images are graphs of holomorphic maps, and they develop a general submersion framework that yields holomorphic retractions, leaves in holomorphic foliations, and complete-intersection results for the image submanifolds. Furthermore, the paper develops a robust theory of non-degeneracy for the associated system of functional equations, including invariance under automorphisms and criteria linking linear degeneracy to FE degeneracy, with implications for the rigidity of unit-disk to bounded symmetric domain isometries. The results illuminate the geometric and analytic structure of holomorphic isometries and connect complex-analytic foliation theory with Bergman-geometry of bounded symmetric domains.

Abstract

We study holomorphic isometries between bounded symmetric domains with respect to the Bergman metrics up to a normalizing constant. In particular, we first consider a holomorphic isometry from the complex unit ball into an irreducible bounded symmetric domain with respect to the Bergman metrics. In this direction, we show that images of (nonempty) affine-linear sections of the complex unit ball must be the intersections of the image of the holomorphic isometry with certain affine-linear subspaces. We also construct a surjective holomorphic submersion from a certain subdomain of the target bounded symmetric domain onto the complex unit ball such that the image of the holomorphic isometry lies inside the subdomain and the holomorphic isometry is a global holomorphic section of the holomorphic submersion. This construction could be generalized to any holomorphic isometry between bounded symmetric domains with respect to the \emph{canonical Kähler metrics}. Using some classical results for complex-analytic subvarieties of Stein manifolds, we have obtained further geometric results for images of such holomorphic isometries.

On geometric properties of holomorphic isometries between bounded symmetric domains

TL;DR

This work analyzes holomorphic isometries from the complex unit ball into irreducible bounded symmetric domains under Bergman-type metrics, revealing a rigid geometric structure of their images. By constructing a bounded balanced convex domain and a surjective holomorphic submersion , the authors show that behaves as a holomorphic section with a holomorphic splitting of tangent spaces, enabling a precise description of how affine-linear sections are mapped. They establish the existence of isometries whose images are graphs of holomorphic maps, and they develop a general submersion framework that yields holomorphic retractions, leaves in holomorphic foliations, and complete-intersection results for the image submanifolds. Furthermore, the paper develops a robust theory of non-degeneracy for the associated system of functional equations, including invariance under automorphisms and criteria linking linear degeneracy to FE degeneracy, with implications for the rigidity of unit-disk to bounded symmetric domain isometries. The results illuminate the geometric and analytic structure of holomorphic isometries and connect complex-analytic foliation theory with Bergman-geometry of bounded symmetric domains.

Abstract

We study holomorphic isometries between bounded symmetric domains with respect to the Bergman metrics up to a normalizing constant. In particular, we first consider a holomorphic isometry from the complex unit ball into an irreducible bounded symmetric domain with respect to the Bergman metrics. In this direction, we show that images of (nonempty) affine-linear sections of the complex unit ball must be the intersections of the image of the holomorphic isometry with certain affine-linear subspaces. We also construct a surjective holomorphic submersion from a certain subdomain of the target bounded symmetric domain onto the complex unit ball such that the image of the holomorphic isometry lies inside the subdomain and the holomorphic isometry is a global holomorphic section of the holomorphic submersion. This construction could be generalized to any holomorphic isometry between bounded symmetric domains with respect to the \emph{canonical Kähler metrics}. Using some classical results for complex-analytic subvarieties of Stein manifolds, we have obtained further geometric results for images of such holomorphic isometries.

Paper Structure

This paper contains 12 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preservation of complex affine-linear sections
  3. Existence of holomorphic isometries whose images are the graphs of certain holomorphic maps
  4. Induced surjective holomorphic submersion
  5. Geometric properties
  6. Holomorphic retractions
  7. On complete intersections
  8. Linear degeneracy of certain holomorphic isometries
  9. Non-degeneracy of holomorphic isometries
  10. Sufficiently non-degeneracy of the system of functional equations
  11. Linearly non-degeneracy
  12. Acknowledgements

Key Result

Proposition 2.1

Let $f:(\mathbb B^n,kg_{\mathbb B^n})$$\to$$(\Omega,g_\Omega)$ be a holomorphic isometric embedding such that $f({\bf 0})={\bf 0}$ and $n\ge 2$, where $\Omega\Subset \mathbb C^N$ is an irreducible bounded symmetric domain in its Harish-Chandra realization and $k$ is an integer satisfying $1\le k \le where ${\bf A}\in M(n-m,n;\mathbb C)$ is a matrix of rank $(n-m)$. In addition, for any $V\in G(m,n

Theorems & Definitions (42)

  • Proposition 2.1: Preservation of complex affine-linear sections
  • proof
  • Claim 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Claim 4.2
  • ...and 32 more