Bergman metrics induced by the ball

TL;DR

This paper characterizes when the Bergman metric of certain complex domains is equivalent to that of a unit ball, establishing rigidity results for strictly pseudoconvex domains and specific classes of Hartogs and egg domains.

Contribution

It provides new rigidity theorems for when Bergman metrics are induced by the ball, including boundary extension conditions and classifications of special domain classes.

Findings

Rigidity for strictly pseudoconvex domains in  with boundary extension

Classification of Hartogs and egg domains with Bergman metric induced by a ball

Explicit algebraic and kernel-based methods to determine metric equivalence

Abstract

We investigate when the Bergman metric of a bounded domain is, up to a constant factor $\lambda$, induced by the Bergman metric of a finite-dimensional unit ball $\mathbb{B}^N$ via a holomorphic isometric immersion. For a strictly pseudoconvex domain in $\mathbb{C}^2$ we prove rigidity: if such an immersion extends smoothly and transversally past the boundary and $(N + 1)/\lambda - 3 \in \mathbb{N}$, then the domain is biholomorphic to the ball. We then consider two broad classes of examples: Hartogs domains over bounded homogeneous bases and egg domains over irreducible symmetric bases, and show that, in finite target dimension, the only members whose (rescaled) Bergman metric is induced by that of a ball are the balls themselves. The proofs combine Calabi's diastasis criterion with explicit Bergman kernel formulas (such as Fefferman's expansion) and algebraic arguments that force arithmetic constraints on the scaling factor. In higher dimensions, the first result follows under a Ramadanov-type assumption.