Shan Tai Chan
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 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 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 $(\Omega,g_\Omega)$.
ArXiv: 1702.01668