On the structure of holomorphic isometric embeddings of complex unit balls into bounded symmetric domains

Shan Tai Chan

Vol. 295 (2018), No. 2, 291–315

DOI: 10.2140/pjm.2018.295.291

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 $\Omega ⋐{ℂ}^N$ 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 $\left(\Omega ,g_{\Omega }\right)$.

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