LAUSR

We study general properties of images of holomorphic isometric embeddings of complex unit balls $${\mathbb {B}}^m$$Bm into irreducible bounded symmetric domains $${\varOmega }$$Ω of rank at least 2. In particular, we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections $${\varLambda }$$Λ of the compact dual $$X_c$$Xc of $${\varOmega }$$Ω. The question naturally arises as to which linear sections $$Z = {\varLambda }\cap {\varOmega }$$Z=Λ∩Ω are actually images of holomorphic isometries of complex unit balls. We study the latter question in the case of bounded symmetric domains $${\varOmega }$$Ω 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 $${\varOmega }$$Ω. 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 $$f: {\mathbb {B}}^m \rightarrow {\varOmega }$$f:Bm→Ω extends to a holomorphic isometric embedding $$f: \mathbb B^{n-1} \rightarrow {\varOmega }$$f:Bn-1→Ω, $$\dim {\varOmega }= n$$dimΩ=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.