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

Autor: Ngaiming Mok, Shan Tai Chan
Rok vydání: 2016
Předmět:
Zdroj: Mathematische Zeitschrift. 286:679-700
ISSN: 1432-1823
0025-5874
DOI: 10.1007/s00209-016-1778-7
Popis: 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.
Databáze: OpenAIRE
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