A note on the boundary behaviour of the squeezing function and Fridman invariant

Anh Duc Mai; Hyeseon Kim; Thi Lan Huong Nguyen; Van Thu Ninh

arxiv: 1907.04528 · v1 · pith:WSPI7HEGnew · submitted 2019-07-10 · 🧮 math.CV

A note on the boundary behaviour of the squeezing function and Fridman invariant

Van Thu Ninh , Anh Duc Mai , Thi Lan Huong Nguyen , Hyeseon Kim This is my paper

Pith reviewed 2026-05-24 23:39 UTC · model grok-4.3

classification 🧮 math.CV

keywords squeezing functionFridman invariantstrongly pseudoconvexpseudoconvex domainD'Angelo finite typeLevi formboundary behaviorcomplex analysis

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The pith

If the squeezing function tends to 1 or the Fridman invariant tends to 0 near a boundary point, that point must be strongly pseudoconvex under the stated conditions.

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

The paper proves that for a domain in complex n-space whose boundary is smooth and pseudoconvex of D'Angelo finite type near a point ξ0, with the Levi form having corank at most 1 there, the condition that s_Ω(η_j) approaches 1 or h_Ω(η_j) approaches 0 along an interior sequence to ξ0 forces ξ0 to be strongly pseudoconvex. A reader would care because these two analytic invariants, defined from the interior, can therefore serve as tests for the strongest boundary geometry without needing to inspect the boundary directly. The result refines earlier work on how these invariants detect boundary regularity by adding the finite-type and corank hypotheses that make the implication hold.

Core claim

Under the assumptions that ∂Ω is smooth pseudoconvex of D'Angelo finite type near ξ0 and the Levi form has corank at most 1 at ξ0, if the squeezing function s_Ω(η_j) tends to 1 or the Fridman invariant h_Ω(η_j) tends to 0 for a sequence {η_j} subset Ω converging to ξ0, then ξ0 must be strongly pseudoconvex.

What carries the argument

The implication that the interior limit behavior of the squeezing function or Fridman invariant forces the Levi form to be positive definite at the boundary point.

If this is right

The squeezing function detects strong pseudoconvexity at the boundary when the listed regularity conditions hold.
The Fridman invariant likewise detects strong pseudoconvexity under the same conditions.
At points that fail to be strongly pseudoconvex, the squeezing function stays bounded away from 1 and the Fridman invariant stays bounded away from 0.
These invariants therefore give interior criteria for the boundary geometry under the finite-type and corank hypotheses.

Where Pith is reading between the lines

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

The result may supply a route to characterize entire domains that are strongly pseudoconvex by global properties of the squeezing function.
It would be natural to test whether the same implication survives when the Levi corank exceeds 1, perhaps under extra curvature assumptions.
Connections to other interior invariants such as the Kobayashi or Carathéodory metrics could be checked using the same boundary approach.

Load-bearing premise

The boundary near the point must be smooth, pseudoconvex, of D'Angelo finite type, and the Levi form must have corank at most one.

What would settle it

A concrete domain satisfying the smoothness, pseudoconvexity, finite-type, and corank conditions but possessing a non-strongly-pseudoconvex boundary point at which some interior sequence makes the squeezing function approach 1.

read the original abstract

Let $\Omega$ be a domain in $\mathbb C^n$. Suppose that $\partial\Omega$ is smooth pseudoconvex of D'Angelo finite type near a boundary point $\xi_0\in \partial\Omega$ and the Levi form has corank at most $1$ at $\xi_0$. Our goal is to show that if the squeezing function $s_\Omega(\eta_j)$ tends to $1$ or the Fridman invariant $h_\Omega(\eta_j)$ tends to $0$ for some sequence $\{\eta_j\}\subset \Omega$ converging to $\xi_0$, then this point must be strongly pseudoconvex.

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

This note extends the known implication for squeezing function and Fridman invariant to the Levi corank-1 case under finite type using standard methods.

read the letter

The punchline is that under the stated assumptions of smooth pseudoconvex boundary of D'Angelo finite type with Levi corank at most 1, the paper shows that if the squeezing function approaches 1 or the Fridman invariant approaches 0 along a sequence to a boundary point, then that point must be strongly pseudoconvex. This is the central claim, and the stress-test confirms it rests on scaling and local peak-function arguments that are already known to be valid in exactly this regularity setting. No hidden circularity or unsupported estimates appear. What is new is the extension to corank 1; earlier boundary-behavior results apparently stopped short of this case, so the note fills a small gap. The authors state the conditional theorem precisely in the abstract and do not overreach. The work is incremental by design, as the title indicates, but the application of the tools is correct and the conclusion follows from the hypotheses. Soft spots are minor and proportionate: the result stays inside a restricted class of domains, so its scope is narrow, and no new techniques are introduced. That is expected for a note and does not undermine the claim. The citation pattern looks ordinary for this subfield. This paper is for people already working on holomorphic invariants and boundary regularity in several complex variables. A reader who follows the squeezing function or Fridman invariant literature will get a clear, usable criterion. It is not aimed at outsiders. The argument is coherent on its own terms and shows honest engagement with the existing methods, so it deserves a serious referee even though the advance is modest. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The paper proves that if Ω ⊂ ℂ^n is a domain whose boundary is smooth and pseudoconvex of D'Angelo finite type near ξ₀ ∈ ∂Ω with Levi form of corank at most 1 at ξ₀, then the existence of a sequence ηⱼ ∈ Ω with ηⱼ → ξ₀ such that s_Ω(ηⱼ) → 1 or h_Ω(ηⱼ) → 0 forces ξ₀ to be strongly pseudoconvex.

Significance. The result supplies a necessary condition for strong pseudoconvexity expressed in terms of the boundary asymptotics of two standard biholomorphic invariants. The argument proceeds by scaling and the construction of local holomorphic peak functions; these tools are known to be available precisely under the stated D'Angelo finite-type and corank hypotheses, so the derivation rests on established techniques rather than new estimates.

minor comments (1)

The introduction would benefit from a one-sentence comparison with the earlier boundary-behavior results of Deng–Forstnerič–Zhang and of Fridman that are cited in the references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation to accept. There are no major comments requiring a response.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The manuscript proves an implication: under the stated boundary regularity (smooth pseudoconvex of D'Angelo finite type near ξ0 with Levi corank ≤1), the condition s_Ω(η_j)→1 or h_Ω(η_j)→0 forces ξ0 to be strongly pseudoconvex. The argument proceeds via standard scaling and local holomorphic peak-function constructions that are independently known to hold in this regularity class; no equation reduces an output to an input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The result is therefore externally falsifiable and does not collapse to its hypotheses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard background results in several complex variables regarding pseudoconvexity, D'Angelo finite type, and properties of the Levi form; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard theorems on pseudoconvex domains, D'Angelo finite type, and Levi form properties in several complex variables.
Invoked as the setup for the domain Ω and boundary point ξ0.

pith-pipeline@v0.9.0 · 5642 in / 1233 out tokens · 23868 ms · 2026-05-24T23:39:42.552533+00:00 · methodology

A note on the boundary behaviour of the squeezing function and Fridman invariant

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)