Holomorphic isometries of Bm into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals

Shan Tai Chan, Ngaiming Mok

Abstract: We study general properties of images of holomorphic isometric embeddings of complex unit balls Bm into irreducible bounded symmetric domains Ω of rank at least 2. In particular, we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections Λ of the compact dual Xc of Ω. The question naturally arises as to which linear sections 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 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 ΩΩ . 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:Bm→Ω extends to a holomorphic isometric embedding f:Bn−1→Ω, 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.

Journal: Mathematische Zeitschrift

DOI: 10.1007/s00209-016-1778-7