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

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 $({\mathbb{B}}^n,k{g}_{{\mathbb{B}}^n})$ 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 $\Omega ⋐{\mathbb{C}}^N$ 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 $({\mathbb{B}}^n,g_{{\mathbb{B}}^n})$ to $(\Omega ,g_{\Omega })$ for $1\le n\le n_0(\Omega )$, where $n_0(\Omega ):=2N-N^{\prime }>1$. 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 $(\Omega ,g_{\Omega })$.

References in corpus (3)

• Volume-preserving maps between Hermitian symmetric spaces of compact type
• Holomorphic maps from the complex unit ball to Type IV classical domains
• Holomorphic isometries from the unit ball into symmetric domains