Some Recent Results on Holomorphic Isometries of the Complex Unit Ball into Bounded Symmetric Domains and Related Problems

Abstract

In his seminal work Calabi established the foundation on the study of holomorphic isometries from a Kahler manifold with real analytic local potential functions into complex space forms, e.g., Fubini-Study spaces. This leads to interior extension results on germs of holomorphic isometries between bounded domains. General results on boundary extension were obtained by Mok under assumptions such as the rationality of Bergman kernels, which applies especially to holomorphic isometries between bounded symmetric domains in their Harish-Chandra realizations. Because of rigidity results in the cases where the holomorphic isometry is defined on an irreducible bounded symmetric domain of rank \({\ge }2\), we focus on holomorphic isometries defined on the complex unit ball \(\mathbb B^n, n \ge 1\). We discuss results on the construction, characterization and classification of holomorphic isometries of the complex unit ball into bounded symmetric domains and more generally into bounded homogeneous domains. Furthermore, in relation to the study of the Hyperbolic Ax-Lindemann Conjecture for not necessarily arithmetic quotients of bounded symmetric domains, such holomorphic isometric embeddings play an important role. We also present some differential-geometric techniques arising from the study of the latter conjecture.