Holomorphic isometries of ${\mathbb {B}}^m$ into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals

Abstract

We study general properties of images of holomorphic isometric embeddings of complex unit balls ${\mathbb {B}}^m$ into irreducible bounded symmetric domains ${\varOmega }$ of rank at least 2. In particular, we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections ${\varLambda }$ of the compact dual $X_c$ of ${\varOmega }$. The question naturally arises as to which linear sections $Z = {\varLambda }\cap {\varOmega }$ are actually images of holomorphic isometries of complex unit balls. We study the latter question in the case of bounded symmetric domains ${\varOmega }$ of type IV, alias Lie balls, i.e., bounded symmetric domains dual to hyperquadrics. We completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains ${\varOmega }$. Especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by Mok (Proc Am Math Soc 144:4515–4525, 2016) from varieties of minimal rational tangents, and that moreover any holomorphic isometric embedding $f: {\mathbb {B}}^m \rightarrow {\varOmega }$ extends to a holomorphic isometric embedding $f: \mathbb B^{n-1} \rightarrow {\varOmega }$, $\dim {\varOmega }= n$. The case of Lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type-IV domain are expected to be more rigid.