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arXiv 1702.01668math.CVmath.DG
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复单位球在有界对称域上的全纯等距嵌入的结构

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

Shan Tai Chan

发表信息：
Pacific J. Math. 295 (2018) 291-315
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摘要：

研究了复单位球$\mathbb B^n$在秩为$\mathbb B^n$的有界对称域上的全纯等距嵌入的一般性质。在第一部分中，我们研究了从$(\mathbb B^n,kg_{\mathbb B^n})$到$(Ω,g_Ω)$的具有非最小等距常数$k$的全纯等距同构，其中$g_D$表示任何秩为$\ge 2$的不可约有界对称整环$D$上的规范Kähler-Einstein度量，其中$D$的极小圆盘具有常高斯曲率$-2$。特别是，结果有关等距嵌入$\mathbb B^n$在$Ω$的尺寸的上限和结构的图像，这样的全纯等距。在第二部分中，我们研究了对于任何秩等于$2$且具有$2N>N'+1$的不可约有界对称整环$Ω\Subset \mathbb C^N$，从$(\mathbb B^n,g_{\mathbb B^n})$到$(Ω,g_Ω)$的全纯等距，其中$N'$是一个整数，使得$ι:X_c\hookrightarrow \mathbb P^{N'}$是最小嵌入（即，第一个典型嵌入）的紧对偶厄米对称空间$X_c$的$Ω$。我们完全分类图像的所有全纯等距从$(\mathbb B^n,g_{\mathbb B^n})$ $(Ω,g_Ω)$为$1\le n \le n_0(Ω)$，其中$n_0(Ω):=2N-N'>1$。特别地，对于$1\le n \le n_0(Ω)-1$，我们证明了从$(\mathbb B^n,g_{\mathbb B^n})$到$(Ω,g_Ω)$的任何全纯同构都可以扩展到从$(\mathbb B^{n_0(Ω)},g_{\mathbb B^{n_0(Ω)}})$到$(Ω,g_Ω)$的某些全纯同构。

英文摘要：

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 $(Ω,g_Ω)$ with non-minimal isometric constants $k$ for any irreducible bounded symmetric domain $Ω$ 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 $Ω$ 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 $(Ω,g_Ω)$ for any irreducible bounded symmetric domains $Ω\Subset \mathbb C^N$ of rank equal to $2$ with $2N>N'+1$, where $N'$ is an integer such that $ι:X_c\hookrightarrow \mathbb P^{N'}$ is the minimal embedding (i.e., the first canonical embedding) of the compact dual Hermitian symmetric space $X_c$ of $Ω$. We completely classify images of all holomorphic isometries from $(\mathbb B^n,g_{\mathbb B^n})$ to $(Ω,g_Ω)$ for $1\le n \le n_0(Ω)$, where $n_0(Ω):=2N-N'>1$. In particular, for $1\le n \le n_0(Ω)-1$ we prove that any holomorphic isometry from $(\mathbb B^n,g_{\mathbb B^n})$ to $(Ω,g_Ω)$ extends to some holomorphic isometry from $(\mathbb B^{n_0(Ω)},g_{\mathbb B^{n_0(Ω)}})$ to $(Ω,g_Ω)$.

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