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arxiv: 2605.17702 · v1 · pith:5N2C3I5Tnew · submitted 2026-05-17 · 🧮 math.CV · math.DG

Boundary Behavior of Bisectional Curvatures for Weighted Bergman Metrics

Sungmin Yoo This is my paper

Pith reviewed 2026-05-19 21:48 UTC · model grok-4.3

classification 🧮 math.CV math.DG
keywords weighted Bergman metricsholomorphic bisectional curvaturestrongly pseudoconvex domainssqueezing functionboundary asymptoticsKähler-Einstein metricextremal functionsL2 projections
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The pith

At strongly pseudoconvex boundary points the holomorphic bisectional curvature of weighted Bergman metrics asymptotically matches that of the unit ball.

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

The paper derives an explicit formula for the holomorphic bisectional curvature of weighted Bergman metrics by characterizing extremal functions through L2-orthogonal projections. It applies the squeezing function to produce quantitative upper and lower bounds on bounded pseudoconvex domains. The central result shows that this curvature approaches the value attained by the unit ball when one approaches a strongly pseudoconvex boundary point. The same technique supplies a unified proof of the known asymptotic behavior for the bisectional curvature of the Kähler-Einstein metric. A reader would care because the boundary geometry of these metrics is thereby reduced to a simple local comparison.

Core claim

By characterizing extremal functions via L²-orthogonal projections, the paper obtains an explicit formula for the weighted bisectional curvature. Applying the squeezing function then yields upper and lower bounds on any bounded pseudoconvex domain. The key discovery is that at strongly pseudoconvex boundary points the holomorphic bisectional curvature of the weighted Bergman metric asymptotically coincides with the bisectional curvature of the unit ball. The same asymptotic statement therefore holds for the Kähler-Einstein metric by a streamlined argument.

What carries the argument

The explicit formula for weighted bisectional curvature obtained by expressing extremal functions as L²-orthogonal projections onto holomorphic sections, together with the squeezing function used to control the resulting bounds.

If this is right

  • The weighted bisectional curvature admits explicit quantitative upper and lower bounds on every bounded pseudoconvex domain.
  • These bounds collapse to a single value at strongly pseudoconvex boundary points, matching the unit ball.
  • The same boundary asymptotics hold for the Kähler-Einstein metric by the identical argument.
  • Local boundary geometry alone determines the curvature limit, independent of global domain features away from the point.

Where Pith is reading between the lines

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

  • The method isolates the contribution of the local boundary geometry from any global holomorphic invariants of the domain.
  • The same projection-plus-squeezing approach may yield asymptotic statements for other curvature tensors on the same class of domains.
  • Results of this type supply a uniform way to compare the boundary behavior of distinct Kähler metrics that share the same underlying complex structure.

Load-bearing premise

The characterization of extremal functions via L2-orthogonal projections holds and produces an explicit formula for the weighted bisectional curvature before the squeezing function is applied.

What would settle it

Compute the limit of the bisectional curvature for a concrete weighted Bergman metric as one approaches a strongly pseudoconvex boundary point on the unit ball or on a smooth strictly convex domain and check whether the limit equals the known constant value for the unit ball.

read the original abstract

This paper investigates the asymptotic boundary behavior of the holomorphic bisectional curvature for weighted Bergman metrics. By characterizing extremal functions via $L^2$-orthogonal projections, we establish an explicit formula for the weighted bisectional curvature. Utilizing the squeezing function, we then obtain quantitative upper and lower bounds for the curvature on bounded pseudoconvex domains. Furthermore, we prove that at strongly pseudoconvex boundary points, the bisectional curvature asymptotically coincides with that of the unit ball. As an application, these results provide a streamlined and unified proof for the known asymptotic behavior of the bisectional curvature of the K\"ahler-Einstein metric.

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

2 major / 2 minor

Summary. The manuscript establishes an explicit formula for the holomorphic bisectional curvature of weighted Bergman metrics on bounded pseudoconvex domains by characterizing extremal functions through L²-orthogonal projections. It then derives quantitative bounds using the squeezing function and proves that at strongly pseudoconvex boundary points, this curvature asymptotically coincides with that of the unit ball. As an application, it offers a unified proof for the asymptotic behavior of the bisectional curvature of the Kähler-Einstein metric.

Significance. If the derivations hold, the paper supplies a streamlined, unified treatment of boundary asymptotics for Bergman-type metrics in several complex variables, extending known results for the unweighted case and the Kähler-Einstein metric via standard tools (L² projections and the squeezing function). This could facilitate further work on curvature questions for weighted metrics.

major comments (2)
  1. [Abstract, first paragraph] Abstract, first paragraph: the claim that the L²-orthogonal projection characterization yields an explicit formula for the weighted bisectional curvature requires verification that the projection remains well-defined and the resulting curvature expression is free of hidden weight-dependent terms that could affect the subsequent asymptotic analysis.
  2. [Section deriving the bounds] Section deriving the bounds: the quantitative upper and lower bounds obtained from the squeezing function must be shown to be independent of post-hoc choices in the weight and to remain tight at strongly pseudoconvex points; otherwise the asymptotic coincidence with the unit ball may not follow directly.
minor comments (2)
  1. [Notation and setup] Clarify the precise definition of the weighted Bergman metric and the associated inner product in the opening sections to avoid ambiguity when passing to the curvature formula.
  2. [Introduction] Add a brief comparison paragraph with existing results on unweighted Bergman curvature asymptotics to highlight the novelty of the weighted extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and have incorporated the requested clarifications into the revised version.

read point-by-point responses

  1. Referee: [Abstract, first paragraph] Abstract, first paragraph: the claim that the L²-orthogonal projection characterization yields an explicit formula for the weighted bisectional curvature requires verification that the projection remains well-defined and the resulting curvature expression is free of hidden weight-dependent terms that could affect the subsequent asymptotic analysis.

    Authors: The weighted Bergman space is a Hilbert space for any positive continuous weight on a bounded domain, so the L²-orthogonal projection onto the holomorphic subspace is well-defined and bounded by standard functional analysis. The explicit curvature formula is obtained by applying this projection to the appropriate test functions; the weight appears explicitly in the inner product but enters the final expression only through the resulting extremal functions. In the boundary asymptotic analysis these weight-dependent contributions are controlled by the squeezing function and vanish in the limit at strongly pseudoconvex points. We have added a short clarifying paragraph after the statement of the formula to make the well-definedness and absence of persistent hidden terms explicit. revision: yes

  2. Referee: [Section deriving the bounds] Section deriving the bounds: the quantitative upper and lower bounds obtained from the squeezing function must be shown to be independent of post-hoc choices in the weight and to remain tight at strongly pseudoconvex points; otherwise the asymptotic coincidence with the unit ball may not follow directly.

    Authors: The squeezing function is a biholomorphic invariant independent of any auxiliary weight. Our upper and lower bounds are expressed solely in terms of the squeezing function evaluated at the interior point; any post-hoc rescaling or normalization of the weight cancels in the ratio that defines the curvature and does not affect the estimates. At strongly pseudoconvex boundary points the squeezing function tends to 1, which forces the bounds to coincide with the constant bisectional curvature of the unit ball. We have inserted a remark in the section on quantitative bounds that explicitly notes this independence and the resulting tightness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard tools

full rationale

The paper's chain begins with the standard L2-orthogonal projection characterization of extremal functions (a field-external fact) to derive an explicit weighted bisectional curvature formula, followed by application of the squeezing function (likewise external) to obtain quantitative bounds and the asymptotic coincidence with the unit ball at strongly pseudoconvex points. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the Kähler-Einstein application is a direct consequence of the new bounds rather than a renaming or imported uniqueness theorem. The derivation is self-contained against external benchmarks in several complex variables.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of extremal functions characterized by L2 projections and on the squeezing function providing quantitative control near the boundary; these are treated as standard tools rather than newly invented.

axioms (2)
  • domain assumption Extremal functions for the weighted Bergman metric can be characterized via L2-orthogonal projections
    Invoked in the first step to establish the explicit formula for the curvature (abstract).
  • domain assumption The squeezing function yields quantitative upper and lower bounds for curvature on bounded pseudoconvex domains
    Used to obtain the bounds before the asymptotic statement.

pith-pipeline@v0.9.0 · 5626 in / 1408 out tokens · 42678 ms · 2026-05-19T21:48:07.542954+00:00 · methodology

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Reference graph

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