On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

Abstract

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F ₁(z) = (0, I _n;m(z)) with the isometric constant equal to (m + 1)/(n + 1) and F ₂(z) = (I _n;m(z), I _n;m(z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.