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Abstract

We study general properties of holomorphic isometric embeddings of complex unit balls
${\mathbb{B}}^{n}$ into bounded symmetric
domains of rank
$\ge 2$.
In the first part, we study holomorphic isometries from
$\left({\mathbb{B}}^{n},k{g}_{{\mathbb{B}}^{n}}\right)$ to
$\left(\Omega ,{g}_{\Omega}\right)$ with nonminimal
isometric constants
$k$
for any irreducible bounded symmetric domain
$\Omega $ of rank
$\ge 2$, where
${g}_{D}$ denotes
the canonical Kähler–Einstein metric on any irreducible bounded symmetric domain
$D$ normalized so that minimal
disks of
$D$ are of constant
Gaussian curvature
$2$.
In particular, results concerning the upper bound of the dimension of isometrically
embedded
${\mathbb{B}}^{n}$
in
$\Omega $
and the structure of the images of such holomorphic isometries are obtained.
In the second part, we study holomorphic isometries from
$\left({\mathbb{B}}^{n},{g}_{{\mathbb{B}}^{n}}\right)$ to
$\left(\Omega ,{g}_{\Omega}\right)$ for any irreducible bounded
symmetric domains
$\Omega \u22d0{\u2102}^{N}$
of rank equal to
$2$
with
$2N>{N}^{\prime}+1$, where
${N}^{\prime}$ is an integer
such that
$\iota :{X}_{c}\hookrightarrow {\mathbb{P}}^{{N}^{\prime}}$ is
the minimal embedding (i.e., the first canonical embedding) of the compact dual Hermitian
symmetric space
${X}_{c}$
of
$\Omega $.
We completely classify images of all holomorphic isometries from
$\left({\mathbb{B}}^{n},{g}_{{\mathbb{B}}^{n}}\right)$ to
$\left(\Omega ,{g}_{\Omega}\right)$ for
$1\le n\le {n}_{0}\left(\Omega \right)$, where
${n}_{0}\left(\Omega \right):=2N{N}^{\prime}>1$. In particular, for
$1\le n\le {n}_{0}\left(\Omega \right)1$ we prove that any
holomorphic isometry from
$\left({\mathbb{B}}^{n},{g}_{{\mathbb{B}}^{n}}\right)$
to
$\left(\Omega ,{g}_{\Omega}\right)$ extends to some
holomorphic isometry from
$\left({\mathbb{B}}^{{n}_{0}\left(\Omega \right)},{g}_{{\mathbb{B}}^{{n}_{0}\left(\Omega \right)}}\right)$
to
$\left(\Omega ,{g}_{\Omega}\right)$.

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Keywords
Bergman metrics, holomorphic isometric embeddings, bounded
symmetric domains, Borel embedding, complex unit balls

Mathematical Subject Classification 2010
Primary: 32M15, 53C55, 53C42

Milestones
Received: 3 February 2017
Revised: 1 November 2017
Accepted: 5 February 2018
Published: 11 April 2018

