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Bergman metrics induced by the ball

Matteo Palmieri

TL;DR

The paper investigates when the Bergman metric of a bounded domain is, up to a scalar, induced by the Bergman metric of a finite-dimensional ball via a holomorphic isometric immersion. It combines Calabi's diastasis criterion with Fefferman's Bergman kernel expansion and holomorphic Nash algebraic methods to obtain arithmetic constraints on the scaling factor and to analyze boundary transversality. The main results establish rigidness: in dimension two a smooth, boundary-transversal immersion forces the domain to be the ball under a specific rationality condition, and for Hartogs- and egg-domain constructions, any finite-target-dimension immersion forces the domain to be a ball; in higher dimensions a Ramadanov-type assumption yields the same rigidity. Together, these results elucidate how kernel asymptotics, boundary geometry, and algebraic constraints rigidify the Bergman-embedding problem into the ball, with implications for complex-analytic and Kähler-geometric structures.

Abstract

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 $\mathbb{B}^N$ 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 $(N + 1)/λ- 3 \in \mathbb{N}$, 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.

Bergman metrics induced by the ball

TL;DR

The paper investigates when the Bergman metric of a bounded domain is, up to a scalar, induced by the Bergman metric of a finite-dimensional ball via a holomorphic isometric immersion. It combines Calabi's diastasis criterion with Fefferman's Bergman kernel expansion and holomorphic Nash algebraic methods to obtain arithmetic constraints on the scaling factor and to analyze boundary transversality. The main results establish rigidness: in dimension two a smooth, boundary-transversal immersion forces the domain to be the ball under a specific rationality condition, and for Hartogs- and egg-domain constructions, any finite-target-dimension immersion forces the domain to be a ball; in higher dimensions a Ramadanov-type assumption yields the same rigidity. Together, these results elucidate how kernel asymptotics, boundary geometry, and algebraic constraints rigidify the Bergman-embedding problem into the ball, with implications for complex-analytic and Kähler-geometric structures.

Abstract

We investigate when the Bergman metric of a bounded domain is, up to a constant factor , induced by the Bergman metric of a finite-dimensional unit ball via a holomorphic isometric immersion. For a strictly pseudoconvex domain in we prove rigidity: if such an immersion extends smoothly and transversally past the boundary and , 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.
Paper Structure (13 sections, 9 theorems, 91 equations)

This paper contains 13 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. The Bergman kernel and metric
  3. Statement of the results
  4. Kähler immersions into Fubini--Study spaces
  5. Transversal extensions of Kähler immersions
  6. Fefferman's expansion
  7. Transversality
  8. Proof of Theorem \ref{['thm1']}
  9. Rigidity of Kähler immersions into the ball
  10. Hartogs-type and egg domains
  11. Holomorphic Nash algebraic functions
  12. Proof of Theorem \ref{['thm2']}
  13. Proof of Theorem \ref{['thm3']}

Key Result

Theorem 1

Let $\Omega \subset \mathbb{C}^2$ be a simply connected strictly pseudoconvex bounded domain with $C^1$-regular boundary. If there is $\lambda > 0$ such that $\frac{N + 1}{\lambda} - 3 \in \mathbb{N}$ and a Kähler immersion $f \colon (\Omega, \lambda g_\Omega) \to (\mathbb{B}^N, g_{\mathbb{B}^N}),\

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Example 2
  • Example 3
  • Theorem 4: Calabi's local criterion
  • Remark 1
  • Remark 2
  • Remark 3
  • ...and 8 more