Vol. 295, No. 2, 2018

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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
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 $\iota :{X}_{c}↪{ℙ}^{{N}^{\prime }}$ 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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