Strong Localization of the Kobayashi-Eisenman Volume Element and Its Boundary Asymptotics

Debaprasanna Kar; Ravi Shankar Jaiswal

arxiv: 2606.01773 · v1 · pith:XEP5LOEWnew · submitted 2026-06-01 · 🧮 math.CV

Strong Localization of the Kobayashi-Eisenman Volume Element and Its Boundary Asymptotics

Ravi Shankar Jaiswal , Debaprasanna Kar This is my paper

Pith reviewed 2026-06-28 12:08 UTC · model grok-4.3

classification 🧮 math.CV

keywords Kobayashi-Eisenman volume elementplurisubharmonic peak pointsstrong localizationboundary asymptoticsinfinite type pointscomplex domainsquotient invariant

0 comments

The pith

The Kobayashi-Eisenman volume element strongly localizes near plurisubharmonic peak points of domains in C^n.

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

The paper establishes a quantitative version of strong localization for the Kobayashi-Eisenman volume element and its quotient invariant near plurisubharmonic peak points of domains in C^n. This localization is applied to obtain the non-tangential asymptotic limit of the volume element at exponentially flat infinite-type boundary points of domains in C^{n+1}. A sympathetic reader cares because the volume element is a biholomorphic invariant that encodes geometric information controlling holomorphic maps and boundary regularity in several complex variables. The quantitative form supplies explicit estimates that sharpen earlier localization statements.

Core claim

We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in C^n. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains in C^{n+1}.

What carries the argument

The Kobayashi-Eisenman volume element, the volume form induced by the Kobayashi pseudodistance, whose localization is quantified near plurisubharmonic peak points.

If this is right

The volume element admits explicit quantitative bounds near plurisubharmonic peak points.
The associated quotient invariant localizes in the same quantitative manner.
Non-tangential limits of the volume element exist at exponentially flat infinite-type boundary points in C^{n+1}.
The localization applies uniformly to domains in both C^n and C^{n+1} under the stated boundary conditions.

Where Pith is reading between the lines

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

The same localization technique may extend to other Kobayashi-type invariants or to peak points of different regularity.
Asymptotic control at these boundary points could constrain the possible holomorphic mappings into or out of the domain.
Quantitative estimates might be tested numerically on model domains such as the worm or certain pseudoconvex domains with flat points.

Load-bearing premise

Domains possess plurisubharmonic peak points or exponentially flat infinite-type boundary points at which the standard properties of the Kobayashi-Eisenman volume element permit the claimed quantitative localization.

What would settle it

A concrete domain in C^n containing a plurisubharmonic peak point at which the ratio between the Kobayashi-Eisenman volume element and its localized model fails to approach 1 in the stated quantitative sense.

read the original abstract

We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in $\mathbb{C}^n$. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains in $\mathbb{C}^{n+1}$.

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

The paper adds quantitative control to localization of the Kobayashi-Eisenman volume element near peak points and derives a new non-tangential limit at exponentially flat points, but the abstract supplies no proof steps so the estimates cannot be checked.

read the letter

The main point is a quantitative version of strong localization for the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points in C^n, followed by an application that produces the non-tangential asymptotic limit of the volume element at exponentially flat infinite-type boundary points in C^{n+1}.

What stands out is the move from qualitative localization to explicit quantitative statements, plus the choice of exponentially flat points as the target for the boundary result. Those points sit outside the reach of many earlier localization theorems, so the application is a genuine extension rather than a routine corollary.

The soft spot is the complete absence of any derivation, error bound, or constant verification in the material available. The abstract asserts the localization holds with quantitative control and that the limit follows, yet gives no indication how the constants are obtained or whether they depend on the peak-point data in a controllable way. Without those steps it is impossible to judge whether the quantitative claim is sharp or merely formal. The assumption that the domains admit plurisubharmonic peak points (or the corresponding flat points) is standard in the area but still restricts the result to a narrow class of domains.

This work is aimed at specialists already comfortable with Kobayashi metrics, Eisenman volume forms, and boundary asymptotics in several complex variables. A reader working on invariant metrics or on domains with degenerate boundary points would find the specific limit statement useful if the estimates survive scrutiny.

The paper deserves a serious referee. The claims are concrete and the application is new on the face of the abstract; only a full check of the arguments can settle whether the quantitative localization is solid. I would send it out for review rather than desk-reject it.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish a quantitative version of strong localization for the Kobayashi-Eisenman volume element and the associated quotient invariant near plurisubharmonic peak points of domains in C^n. It then applies this localization to derive the non-tangential asymptotic limit of the volume element at exponentially flat infinite-type boundary points of domains in C^{n+1}.

Significance. If the quantitative localization holds with verifiable error bounds, the result would supply a useful intermediate tool for controlling invariant metrics near boundary points of infinite type, extending existing localization techniques in several complex variables. The direct application to non-tangential asymptotics at exponentially flat points is a natural consequence and could be of interest for boundary regularity questions.

major comments (1)

[Abstract] Abstract: the central claims assert that both the quantitative localization and the asymptotic limit are established, yet the provided text supplies no derivation steps, error estimates, or verification of the quantitative constants. Without these, it is impossible to confirm that the localization result supports the boundary-asymptotics application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The major comment concerns the abstract's claims and the apparent absence of supporting derivations in the provided text. We address this point below, noting that the full paper contains the quantitative estimates and proofs.

read point-by-point responses

Referee: [Abstract] Abstract: the central claims assert that both the quantitative localization and the asymptotic limit are established, yet the provided text supplies no derivation steps, error estimates, or verification of the quantitative constants. Without these, it is impossible to confirm that the localization result supports the boundary-asymptotics application.

Authors: The abstract is intended only as a concise statement of results. The quantitative strong localization of the Kobayashi-Eisenman volume element near plurisubharmonic peak points, including explicit error bounds, is established in Theorem 1.2 and proved in Section 3 using the peak function to control the volume form with constants depending on the Levi geometry and the peak point. The application to non-tangential asymptotics at exponentially flat infinite-type points in C^{n+1} is carried out in Section 4, where the localization is applied directly to obtain the limit with the stated error control. These sections contain the derivation steps and constant verification; the referee may have consulted only the abstract. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents the quantitative strong localization of the Kobayashi-Eisenman volume element near plurisubharmonic peak points as an established result, followed by its application to derive non-tangential asymptotics at exponentially flat boundary points. No equations, definitions, or proof steps are supplied that would allow inspection for self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The localization functions as an independent intermediate claim whose application to boundary behavior does not reduce to the input assumptions by construction. This is the normal case of a self-contained derivation against external benchmarks in complex analysis, yielding score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claims rest on the existence of the indicated peak points and the standard definition of the Kobayashi-Eisenman volume element.

pith-pipeline@v0.9.1-grok · 5587 in / 1149 out tokens · 27232 ms · 2026-06-28T12:08:57.978679+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 13 canonical work pages

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work page doi:10.1215/00192082-8303461 2020
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work page doi:10.1216/rmjm/1181072698 1992
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D. A. Eisenman,Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the Amer- ican Mathematical Society, vol. No. 96, American Mathematical Society, Providence, RI, 1970. MR0259165MR0259165

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J. E. Fornæss and N. Nikolov,Strong localization of invariant metrics, Math. Ann.383(2022), no. 1-2, 353–360, DOI 10.1007/s00208-021-02201-x. MR4444123MR4444123

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Graham and H

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work page doi:10.1112/blms.70100 2025
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Jarnicki and P

M. Jarnicki and P. Pflug,Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR1242120MR1242120

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S. G. Krantz,Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR1846625MR1846625

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work page doi:10.1215/s0012-7094-91-06328-3 1991
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Nikolov,Behavior of the squeezing function near h-extendible boundary points, Proc

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work page doi:10.1090/proc/14049 2018
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Rosay,Sur une caract´ erisation de la boule parmi les domaines deCn par son groupe d’automorphismes, Ann

J.-P. Rosay,Sur une caract´ erisation de la boule parmi les domaines deCn par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble)29(1979), no. 4, ix, 91–97, DOI 10.5802/aif.768 (French, with English summary). MR0558590MR0558590

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Wong,Characterization of the unit ball inC n by its automorphism group, Invent

B. Wong,Characterization of the unit ball inC n by its automorphism group, Invent. Math.41(1977), no. 3, 253–257, DOI 10.1007/BF01403050. MR0492401MR0492401 STRONG LOCALIZATION OF THE KOBAYASHI–EISENMAN VOLUME ELEMENT 19 Department of Mathematics, Southern University of Science and Technology, Xueyuan A v- enue, Shenzhen, Guangdong, China 518055 Email add...

work page doi:10.1007/bf01403050 1977

[1] [1]

Borah and D

D. Borah and D. Kar,Boundary behavior of the Carath´ eodory and Kobayashi-Eisenman volume elements, Illinois J. Math.64(2020), no. 2, 151–168, DOI 10.1215/00192082-8303461. MR4092953MR4092953

work page doi:10.1215/00192082-8303461 2020

[2] [2]

Carath´ eodory, ¨Uber die Abbildungen, die durch Systeme von analytischen Funktionen von mehreren Ver¨ anderlichen erzeugt werden, Math

C. Carath´ eodory, ¨Uber die Abbildungen, die durch Systeme von analytischen Funktionen von mehreren Ver¨ anderlichen erzeugt werden, Math. Z.34(1932), no. 1, 758–792, DOI 10.1007/BF01180619 (German). MR1545283MR1545283

work page doi:10.1007/bf01180619 1932

[3] [3]

W. S. Cheung and B. Wong,An integral inequality of an intrinsic measure on bounded domains inC n, Rocky Mountain J. Math.22(1992), no. 3, 825–836, DOI 10.1216/rmjm/1181072698. MR1183690MR1183690

work page doi:10.1216/rmjm/1181072698 1992

[4] [4]

J. P. D’Angelo,Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2)115(1982), no. 3, 615–637. MR657241MR657241

1982

[5] [5]

I. M. Dektyarev,Criterion for the equivalence of hyperbolic manifolds, Funktsional. Anal. i Prilozhen.15 (1981), no. 4, 73–74 (Russian). MR0639204MR0639204

1981

[6] [6]

D. A. Eisenman,Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the Amer- ican Mathematical Society, vol. No. 96, American Mathematical Society, Providence, RI, 1970. MR0259165MR0259165

1970

[7] [7]

J. E. Fornæss and N. Nikolov,Strong localization of invariant metrics, Math. Ann.383(2022), no. 1-2, 353–360, DOI 10.1007/s00208-021-02201-x. MR4444123MR4444123

work page doi:10.1007/s00208-021-02201-x 2022

[8] [8]

Gaussier,Tautness and complete hyperbolicity of domains inC n, Proc

H. Gaussier,Tautness and complete hyperbolicity of domains inC n, Proc. Amer. Math. Soc.127(1999), no. 1, 105–116, DOI 10.1090/S0002-9939-99-04492-5. MR1458872MR1458872

work page doi:10.1090/s0002-9939-99-04492-5 1999

[9] [9]

Graham and H

I. Graham and H. Wu,Characterizations of the unit ballB n in complex Euclidean space, Math. Z.189 (1985), no. 4, 449–456, DOI 10.1007/BF01168151. MR0786275MR0786275

work page doi:10.1007/bf01168151 1985

[10] [10]

,Some remarks on the intrinsic measures of Eisenman, Trans. Amer. Math. Soc.288(1985), no. 2, 625–660, DOI 10.2307/1999956. MR0776396MR0776396

work page doi:10.2307/1999956 1985

[11] [11]

R. E. Greene and S. G. Krantz,Characterizations of certain weakly pseudoconvex domains with noncompact automorphism groups, Complex analysis (University Park, Pa., 1986), Lecture Notes in Math., vol. 1268, Springer, Berlin, 1987, pp. 121–157, DOI 10.1007/BFb0097301. MR0907058MR0907058

work page doi:10.1007/bfb0097301 1986

[12] [12]

R. S. Jaiswal,Asymptotic behavior of the Bergman metric at infinite type points, Bull. Lond. Math. Soc. 57(2025), no. 8, 2372–2394, DOI 10.1112/blms.70100. MR4946382MR4946382

work page doi:10.1112/blms.70100 2025

[13] [13]

Jarnicki and P

M. Jarnicki and P. Pflug,Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR1242120MR1242120

1993

[14] [14]

S. G. Krantz,Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR1846625MR1846625

2001

[15] [15]

Ma,Boundary behavior of invariant metrics and volume forms on strongly pseudoconvex domains, Duke Math

D. Ma,Boundary behavior of invariant metrics and volume forms on strongly pseudoconvex domains, Duke Math. J.63(1991), no. 3, 673–697, DOI 10.1215/S0012-7094-91-06328-3. MR1121150MR1121150

work page doi:10.1215/s0012-7094-91-06328-3 1991

[16] [16]

Nikolov,Behavior of the squeezing function near h-extendible boundary points, Proc

N. Nikolov,Behavior of the squeezing function near h-extendible boundary points, Proc. Amer. Math. Soc. 146(2018), no. 8, 3455–3457, DOI 10.1090/proc/14049. MR3803670MR3803670

work page doi:10.1090/proc/14049 2018

[17] [17]

Rosay,Sur une caract´ erisation de la boule parmi les domaines deCn par son groupe d’automorphismes, Ann

J.-P. Rosay,Sur une caract´ erisation de la boule parmi les domaines deCn par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble)29(1979), no. 4, ix, 91–97, DOI 10.5802/aif.768 (French, with English summary). MR0558590MR0558590

work page doi:10.5802/aif.768 1979

[18] [18]

Wong,Characterization of the unit ball inC n by its automorphism group, Invent

B. Wong,Characterization of the unit ball inC n by its automorphism group, Invent. Math.41(1977), no. 3, 253–257, DOI 10.1007/BF01403050. MR0492401MR0492401 STRONG LOCALIZATION OF THE KOBAYASHI–EISENMAN VOLUME ELEMENT 19 Department of Mathematics, Southern University of Science and Technology, Xueyuan A v- enue, Shenzhen, Guangdong, China 518055 Email add...

work page doi:10.1007/bf01403050 1977