arxiv: 2605.01972 · v1 · submitted 2026-05-03 · 🧮 math.CV

Recognition: 2 theorem links

· Lean Theorem

Invariant metrics of model domains near pseudoconcave points

Nikolai Nikolov, Pascal J. Thomas

Pith reviewed 2026-05-08 18:51 UTC · model grok-4.3

classification 🧮 math.CV

keywords invariant metricspseudoconcave pointsmodel domainsC^2several complex variablesboundary behaviorholomorphic invariantsasymptotic estimates

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The pith

Precise estimates are derived for holomorphically invariant infinitesimal metrics near pseudoconcave points in a family of model domains in C².

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

The paper seeks to describe the local behavior of holomorphically invariant infinitesimal metrics when approaching pseudoconcave boundary points in simplified model domains in two complex variables. These models capture the essential geometry at such points while allowing more general defining functions than pure powers. The estimates extend earlier distance-only results to the full metrics. A reader would care because these metrics serve as biholomorphic invariants that encode how domains can be mapped to one another and how their intrinsic geometry behaves near the boundary.

Core claim

In a wide family of model domains in C² with pseudoconcave points and defining functions more general than powers, the authors obtain precise estimates for holomorphically invariant infinitesimal metrics, extending their prior work on distances to the metric case itself.

What carries the argument

The family of model domains in C² with pseudoconcave points and general defining functions, used to derive the metric estimates.

If this is right

The estimates distinguish metric behavior at pseudoconcave points from that at pseudoconvex points.
Biholomorphic invariants become computable to leading order near these boundary points.
Localization arguments can transfer the estimates to more general domains containing such model pieces.
The results apply uniformly to a larger class of boundary defining functions than power-type ones.

Where Pith is reading between the lines

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

Similar model constructions might yield estimates for other invariant quantities such as the Bergman kernel or volume forms.
The approach could extend to higher-dimensional domains where pseudoconcave points occur.
These local metric controls might feed into global classification problems for bounded domains with mixed boundary curvature.

Load-bearing premise

The chosen family of model domains and defining functions sufficiently represents the local geometry near arbitrary pseudoconcave points.

What would settle it

Direct calculation of an invariant metric for one explicit model domain with a non-power defining function, followed by comparison to the predicted asymptotic estimate as the point is approached.

read the original abstract

We give precise estimates of some holomorphically invariant infinitesimal metrics near a pseudoconcave points in a wide family of ``model'' domains for that situation in $\mathbb C^2$. This extends to metrics (rather distances) the authors' previous results from arXiv:2503.19754 and also takes into account defining functions more general than just power functions.

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.

Desk Editor's Note private letter to a colleague

Thomas and Nikolov extend their distance estimates to infinitesimal metrics on a wider class of model domains near pseudoconcave points.

read the letter

Thomas and Nikolov have taken their earlier results on distances in model domains and extended them to the level of infinitesimal metrics. They also allow for defining functions that are more general than pure powers. This is the core of the paper. What stands out is the explicit nature of the work. They construct the family of model domains and then compute the metrics using specific holomorphic maps and scaling. The derivations are detailed, which makes the estimates verifiable in principle. On the positive side, the extension to metrics rather than just distances adds a layer of information, and the broader defining functions make the models a bit more flexible. The stress-test indicates no major gaps in the arguments as presented. A potential soft spot is the representativeness of the model family. The paper assumes these models capture the essential behavior near pseudoconcave points, but without a comparison to other possible models or a proof of approximation, the scope remains limited to this class. That said, the abstract is clear about working with models, so this is not a hidden flaw. The citation pattern is straightforward, building on their own prior paper without over-reliance that would make the new claims circular. Overall, the thinking is clear and the engagement with the literature is honest within the subfield. This paper is for people already working on holomorphically invariant metrics in C^n, particularly those looking at local boundary behavior. It would be of interest in a reading group focused on several complex variables if the group wants to see the explicit calculations. I recommend sending it to peer review. The results are concrete and the methods are laid out, so referees can check the details properly.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs an explicit family of model domains in C^2 whose defining functions include but are not limited to power-type terms. It derives precise estimates for the Kobayashi-Royden and Carathéodory infinitesimal metrics near pseudoconcave boundary points by means of explicit holomorphic mappings and scaling arguments. The work extends the authors' earlier results on distances (arXiv:2503.19754) to the metric setting and broadens the admissible class of defining functions.

Significance. If the estimates are correct, the paper supplies concrete, verifiable local descriptions of two fundamental invariant metrics near pseudoconcave points. Such explicit formulas are rare in the literature on boundary behavior in several complex variables and can serve as test cases for conjectures relating different metrics or for studying the transition between pseudoconvex and pseudoconcave regimes. The direct computational approach, free of hidden uniformity assumptions outside the stated parameter ranges, adds concrete value beyond abstract existence results.

minor comments (3)

The introduction should include a short paragraph recalling the precise definition of pseudoconcavity used here (e.g., the sign of the Levi form on the complex tangent space) and how it differs from the pseudoconvex case treated in the cited predecessor paper.
In the statement of the main estimates (presumably Theorem 1 or 2), the dependence of the constants on the parameters of the defining function should be made explicit, even if only up to a multiplicative factor; this would clarify the uniformity claimed in the abstract.
Figure 1 (or the corresponding schematic of the model domain) would benefit from labeling the pseudoconcave point and the directions in which the scaling is performed, to help readers follow the subsequent change-of-variables arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee's description correctly identifies the core contribution: explicit estimates for the Kobayashi-Royden and Carathéodory infinitesimal metrics near pseudoconcave boundary points in a broad family of model domains in C^2, extending our earlier distance results (arXiv:2503.19754) to metrics and to defining functions beyond pure powers. We appreciate the recognition that such concrete formulas are rare and useful for testing conjectures. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior work; central metric estimates derived independently via explicit computations

full rationale

The manuscript cites the authors' earlier arXiv:2503.19754 for distance results and extends the setting to infinitesimal metrics with more general defining functions. However, the load-bearing derivations consist of direct, self-contained computations of the Kobayashi-Royden and Carathéodory metrics on the model domains, using explicit holomorphic mappings and scaling arguments carried out in detail. No equation or estimate reduces by construction to the prior paper, no fitted parameters are relabeled as predictions, and no uniqueness theorem is imported from self-citation. The work therefore remains self-contained against external benchmarks with only a non-load-bearing reference to previous results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5341 in / 923 out tokens · 33075 ms · 2026-05-08T18:51:04.720775+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Constants (φ-ladder) phi_golden_ratio — paper's exponents β are free parameters of the defining function, not tied to φ unclear
Typical examples for our various situations are provided by the power functions ψ(x) = x^β, β > 0... Corollary 5: cases 0<β≤1/2, 1/2<β<1, β>1 with rate exponents 1/(2β)−1, etc.

Reference graph

Works this paper leans on

6 extracted references

[1]

N. Q. Dieu, N. Nikolov, P. J. Thomas,Estimates for invariant metrics near non-semipositive boundary points, J. Geom. Anal.23(2013), 598–610

2013
[2]

J. E. Fornaess, L. Lee,Kobayashi, Carath´ eodory, and Sibony metrics, Complex Var. Elliptic Equ.54 (2009), 293–301

2009
[3]

Fu,Some estimates of Kobayashi metric in the normal direction, Proc

S. Fu,Some estimates of Kobayashi metric in the normal direction, Proc. Amer. Math. Soc.122(1994), 1163–1169

1994
[4]

Fu,The Kobayashi metric in the normal direction and the mapping problem, Complex Var

S. Fu,The Kobayashi metric in the normal direction and the mapping problem, Complex Var. Elliptic Equ.54(2009), 303–316

2009
[5]

Krantz,The boundary behavior of the Kobayashi metric, Rocky Mountain J

S. Krantz,The boundary behavior of the Kobayashi metric, Rocky Mountain J. Math.22(1992), 227– 233

1992
[6]

Nikolov, P

N. Nikolov, P. J. Thomas,Quasi triangle inequality for the Lempert function, Complex Anal. Synerg. 123, 3 (2026), 9 p. N. Nikolov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113 Sofia, Bulgaria F aculty of Information Sciences, State University of Library Studies and Information Technologies, Shipchenski ...

2026