2025/10/20 by Matteo Palmieri, Palmieri, Matteo
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2510.17618
openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
We investigate when the Bergman metric of a bounded domain is, up to a constant factor λ, induced by the Bergman metric of a finite-dimensional unit ball \mathbbBN via a holomorphic isometric immersion. For a strictly pseudoconvex domain in ℂ2 we prove rigidity: if such an immersion extends smoothly and transversally past the boundary and (N + 1)/λ- 3 ∈ 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.