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

Abstract

We study general properties of holomorphic isometric embeddings of complex unit balls B(double-struck)n into bounded symmetric domains of rank ≥ 2. In the first part, we study holomorphic isometries from (B(double-struck)n, kgB(double-struck)n) to (Ω, gΩ) with nonminimal isometric constants k for any irreducible bounded symmetric domain Ω of rank ≥ 2, where gD 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 B(double-struck)n in Ω and the structure of the images of such holomorphic isometries are obtained. In the second part, we study holomorphic isometries from (B(double-struck)n, gB(double-struck)n ) to (Ω, gΩ) for any irreducible bounded symmetric domains Ω (double subset) N of rank equal to 2 with 2N > N'+1, where N' is an integer such that ℓ: Xc, (right arrow, hooked)N' is the minimal embedding (i.e., the first canonical embedding) of the compact dual Hermitian symmetric space Xc of Ω. We completely classify images of all holomorphic isometries from (B(double-struck)n, gB(double-struck)n ) to (Ω, gΩ) for 1 ≤ n ≤ n0(Ω), where n0(Ω):=: 2N - N' > 1. In particular, for 1 ≤ n ≤ n0(Ω)-1 we prove that any holomorphic isometry from (B(double-struck)n, gB(double-struck)n ) to (Ω, gΩ) extends to some holomorphic isometry from (B(double-struck)n0(Ω), gB(double-struck)n0(Ω) to (Ω, gΩ). © 2018 Mathematical Sciences Publishers.