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arxiv: 2502.15089 · v2 · submitted 2025-02-20 · 🧮 math.CV

Domains with Bergman metrics of constant curvature and Bergman-negligible subsets

Peter Ebenfelt , John N. Treuer , Ming Xiao This is my paper

Pith reviewed 2026-05-23 01:42 UTC · model grok-4.3

classification 🧮 math.CV
keywords Bergman metricholomorphic sectional curvatureconstant curvatureunit ballbiholomorphismL2 holomorphic functionsmeasure zero setscomplex analysis
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The pith

A bounded domain whose Bergman metric has constant negative curvature must be the unit ball minus a measure-zero set.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers bounded domains in complex n-space where the Bergman metric has holomorphic sectional curvature equal to a fixed negative constant τ. It establishes that any such domain is biholomorphic to the unit ball with a relatively closed measure-zero subset removed. Square-integrable holomorphic functions defined on this modified domain extend to the full unit ball. The constant τ is therefore forced to equal the curvature value already known for the ball itself. This extends an earlier result of Lu and yields applications to rigidity statements of Wong and Rosay type.

Core claim

If a bounded domain D in C^n carries a Bergman metric whose holomorphic sectional curvature is the negative constant τ, then D is biholomorphic to a domain Ω obtained by deleting a relatively closed measure-zero set from the unit ball, and every L²-holomorphic function on Ω extends to an L²-holomorphic function on the ball; consequently τ equals the holomorphic sectional curvature of the ball.

What carries the argument

Biholomorphism to the unit ball minus a relatively closed measure-zero set, together with L²-extension of holomorphic functions across that set.

If this is right

  • The curvature constant τ must coincide with that of the unit ball.
  • All L²-holomorphic functions on the punctured domain extend across the removed set to the full ball.
  • The result generalizes Lu's classical theorem on domains with constant Bergman curvature.
  • The same conclusion supplies extensions of Wong-Rosay rigidity theorems to this curvature setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The measure-zero removable set suggests that holomorphic L² functions are insensitive to sets of this size in the Bergman space.
  • One could ask whether the same conclusion holds when the curvature is allowed to be constant only outside a thin set rather than everywhere.
  • The biholomorphic rigidity may connect to questions of removable singularities for square-integrable holomorphic functions on bounded domains.

Load-bearing premise

The domain is bounded in C^n and its Bergman metric has constant negative holomorphic sectional curvature.

What would settle it

Exhibit a bounded domain in C^n whose Bergman metric has constant negative holomorphic sectional curvature, yet the domain fails to be biholomorphic to the unit ball minus any relatively closed measure-zero set, or the constant differs from the ball's curvature value.

read the original abstract

Let $D$ be a bounded domain in $\mathbb{C}^n$. Suppose the holomorphic sectional curvature of its Bergman metric equals a negative constant $\tau$. We show that $D$ is biholomorphic to a domain $\Omega$ equal to the unit ball in $\mathbb{C}^n$ less a relatively closed set of measure zero, and that all $L^2$-holomorphic functions on $\Omega$ extend to $L^2$-holomorphic functions on the ball. Consequently, $\tau$ must equal the holomorphic sectional curvature of the unit ball. This generalizes a classical theorem of Lu. Some applications of the theorem, especially in extending classical work of Wong and Rosay, are also presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that if D is a bounded domain in C^n whose Bergman metric has constant negative holomorphic sectional curvature τ, then D is biholomorphic to the unit ball minus a relatively closed measure-zero set Ω, all L^2-holomorphic functions on Ω extend to the ball, and therefore τ must equal the holomorphic sectional curvature of the unit ball. This generalizes Lu's theorem, with additional applications to results of Wong and Rosay.

Significance. If the central rigidity argument holds, the result strengthens classification theorems for domains admitting constant-curvature Bergman metrics and supplies a clean extension property for L^2-holomorphic functions. The direct reduction to the ball (up to negligible sets) and the consequent curvature identification constitute a clean generalization of Lu's theorem; the applications to Wong-Rosay-type boundary rigidity are a natural and useful corollary.

minor comments (3)
  1. The abstract states that Ω equals the unit ball less a relatively closed set of measure zero, but does not indicate whether the biholomorphism is required to preserve the Bergman metric or only the complex structure; a clarifying sentence would help.
  2. The statement that 'all L^2-holomorphic functions on Ω extend' should specify the precise L^2 space (with respect to Lebesgue measure on the ball) to avoid ambiguity with the weighted spaces that sometimes appear in Bergman theory.
  3. The applications section is mentioned but not summarized; a one-sentence indication of the Wong-Rosay extension would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation of minor revision. The referee's description correctly identifies the generalization of Lu's theorem and the applications to Wong-Rosay type results.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a rigidity theorem generalizing Lu's classical result: a bounded domain whose Bergman metric has constant negative holomorphic sectional curvature τ must be biholomorphic to the ball minus a measure-zero set, with L²-holomorphic functions extending, forcing τ to match the ball's curvature. The derivation relies on direct analytic arguments from the curvature assumption and boundedness, without self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claim to its inputs. The logic chain (curvature condition → biholomorphism + extension property → curvature equality) is self-contained against external benchmarks and contains no quoted steps that reduce by construction to the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the existence and properties of the Bergman metric on bounded domains together with standard facts from complex differential geometry; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption The Bergman metric is a well-defined Kähler metric on any bounded domain in C^n whose holomorphic sectional curvature can be computed pointwise.
    Invoked at the outset when the constant-curvature hypothesis is stated.
  • standard math Standard results on L2 holomorphic functions and biholomorphic invariance of the Bergman metric hold.
    Used implicitly to conclude function extension and curvature equality.

pith-pipeline@v0.9.0 · 5649 in / 1276 out tokens · 43029 ms · 2026-05-23T01:42:43.281869+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Abundance of Bergman metrics with constant positive holomorphic sectional curvature

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    For every pair of positive integers (m, n) with n >= 2 there exists an R-parameter family of mutually Bergman-inequivalent Reinhardt domains in C^n whose Bergman metrics are locally isometric to m times the Fubini-Stu...

  2. Local Rigidity of the Bergman Metric and of the K\"ahler Carath\'eodory Metric

    math.CV 2024-08 unverdicted novelty 6.0

    Proves that local Kählerness of the Carathéodory metric near the boundary on strictly pseudoconvex domains implies biholomorphism to the ball, plus rigidity for constant-curvature Bergman metrics.

Reference graph

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