Vol. 295, No. 2, 2018

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

Shan Tai Chan

Vol. 295 (2018), No. 2, 291–315

We study general properties of holomorphic isometric embeddings of complex unit balls Bn into bounded symmetric domains of rank 2. In the first part, we study holomorphic isometries from (Bn ,kgBn) 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 Bn in Ω and the structure of the images of such holomorphic isometries are obtained.

In the second part, we study holomorphic isometries from (Bn ,gBn) to (Ω,gΩ) for any irreducible bounded symmetric domains Ω N of rank equal to 2 with 2N > N + 1, where N is an integer such that ι : XcN 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 (Bn ,gBn) 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 (Bn ,gBn) to (Ω,gΩ) extends to some holomorphic isometry from (Bn0(Ω) ,gBn 0(Ω)) to (Ω,gΩ).

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Bergman metrics, holomorphic isometric embeddings, bounded symmetric domains, Borel embedding, complex unit balls
Mathematical Subject Classification 2010
Primary: 32M15, 53C55, 53C42
Received: 3 February 2017
Revised: 1 November 2017
Accepted: 5 February 2018
Published: 11 April 2018
Shan Tai Chan
Department of Mathematics
Syracuse University
Syracuse, NY
United States