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

## TL;DR

This paper investigates the structure and classification of holomorphic isometric embeddings of complex unit balls into bounded symmetric domains, providing bounds on dimensions, image structures, and extension properties of such embeddings.

## Contribution

It offers new bounds on the dimension of isometrically embedded balls and classifies embeddings for certain ranks, including extension results for these embeddings.

## Key findings

- Upper bounds on the dimension of embedded balls
- Complete classification of embeddings for certain ranks
- Extension of embeddings to higher dimensions

## 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 $(\mathbb B^n,kg_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ with non-minimal isometric constants $k$ for any irreducible bounded symmetric domain $\Omega$ of rank $\ge 2$, where $g_D$ denotes the canonical K\"ahler-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 were obtained. In the second part, we study holomorphic isometries from $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ for any irreducible bounded symmetric domains $\Omega\Subset \mathbb C^N$ of rank equal to $2$ with $2N>N'+1$, where $N'$ is an integer such that $\iota:X_c\hookrightarrow \mathbb P^{N'}$ 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 $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ for $1\le n \le n_0(\Omega)$, where $n_0(\Omega):=2N-N'>1$. In particular, for $1\le n \le n_0(\Omega)-1$ we prove that any holomorphic isometry from $(\mathbb B^n,g_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ extends to some holomorphic isometry from $(\mathbb B^{n_0(\Omega)},g_{\mathbb B^{n_0(\Omega)}})$ to $(\Omega,g_\Omega)$.