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arxiv: 2606.13569 · v1 · pith:7RAUAE7Ynew · submitted 2026-06-11 · 🧮 math.CV · math.DG

K\"ahler Hyperbolicity Modulus for Simply-connected K\"ahler Hyperbolic manifolds

Young-Jun Choi , Kang-Hyurk Lee This is my paper

Pith reviewed 2026-06-27 04:58 UTC · model grok-4.3

classification 🧮 math.CV math.DG
keywords Kähler hyperbolicity modulusplurisubharmonic functionhyperconvex domainsstrongly pseudoconvex domainsKähler-Einstein metricsBergman metricsbounded symmetric domains
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The pith

The Kähler hyperbolicity modulus admits a lower bound from the boundary behavior of the gradient length of a plurisubharmonic function.

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

The paper aims to find quantitative lower bounds on the Kähler hyperbolicity modulus for complete Kähler manifolds that are simply connected and hyperbolic, with emphasis on hyperconvex domains and bounded strongly pseudoconvex domains. It shows that the modulus is controlled from below by the limiting behavior of the gradient length of a plurisubharmonic function as one approaches the boundary. A reader would care because the modulus encodes a global measure of hyperbolicity that governs volume growth and curvature negativity, and explicit bounds make this quantity computable rather than abstract. The work then uses the bound to evaluate the modulus on bounded symmetric domains and to obtain estimates when the manifold carries a Kähler-Einstein or Bergman metric.

Core claim

The central claim is that on hyperconvex domains and on bounded strongly pseudoconvex domains the Kähler hyperbolicity modulus is bounded below by an expression that depends only on the boundary asymptotics of the gradient length of a plurisubharmonic function; the same bound yields the exact value of the modulus for all bounded symmetric domains and supplies positive lower bounds for the same modulus when the domain is equipped with its Kähler-Einstein metric or its Bergman metric.

What carries the argument

The Kähler hyperbolicity modulus, defined via the infimum of constants controlling the Kähler form on the universal cover.

If this is right

  • The modulus equals a concrete number for every bounded symmetric domain.
  • Positive lower bounds hold for bounded strongly pseudoconvex domains carrying Kähler-Einstein metrics.
  • Positive lower bounds also hold when the same domains carry Bergman metrics.
  • The modulus is therefore determined, at least from below, by local boundary data of one auxiliary function.

Where Pith is reading between the lines

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

  • The same gradient-length technique may produce lower bounds on other complete Kähler manifolds once a suitable plurisubharmonic function is exhibited.
  • One could check the sharpness of the bound by comparing it with the modulus computed directly on the ball or on the polydisk.
  • The result supplies a concrete way to test whether a given Kähler-Einstein metric on a strongly pseudoconvex domain saturates the hyperbolicity modulus.

Load-bearing premise

The manifold admits a plurisubharmonic function whose gradient length remains controlled near the boundary in a way that can be compared directly with the modulus.

What would settle it

An explicit hyperconvex domain or bounded strongly pseudoconvex domain for which the numerically computed Kähler hyperbolicity modulus falls below the lower bound predicted by the gradient-length expression.

read the original abstract

This paper investigates the K\"ahler hyperbolicity modulus on complete K\"ahler manifolds, with a particular focus on hyperconvex domains and bounded strongly pseudoconvex domains. Our main result establishes a lower bound for the K\"ahler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function. As applications, we compute the K\"ahler hyperbolicity modulus for bounded symmetric domains. Furthermore, we obtain lower bounds for the K\"ahler hyperbolicity modulus on bounded strongly pseudoconvex domains equipped with K\"ahler-Einstein metrics or Bergman metrics.

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 manuscript studies the Kähler hyperbolicity modulus on complete Kähler manifolds, with emphasis on hyperconvex domains and bounded strongly pseudoconvex domains. Its central claim is a lower bound on the modulus expressed in terms of the boundary behavior of | abla u| for a plurisubharmonic exhaustion function u. Applications include explicit evaluation of the modulus on bounded symmetric domains and lower bounds for the same quantity on strongly pseudoconvex domains equipped with Kähler-Einstein or Bergman metrics.

Significance. If the stated lower bound is established, the work supplies a concrete link between the hyperbolicity modulus and standard exhaustion functions, which may facilitate explicit computations on symmetric domains and estimates on pseudoconvex domains. The applications to Kähler-Einstein and Bergman metrics are potentially useful for relating hyperbolicity invariants to well-studied complete metrics.

minor comments (3)
  1. [Abstract] The abstract and introduction should include a precise definition or reference to the Kähler hyperbolicity modulus (including the role of the simply-connected assumption) so that the lower-bound statement can be read without external lookup.
  2. Clarify the precise regularity and boundary conditions imposed on the plurisubharmonic function u in the main theorem; the current phrasing leaves open whether the bound holds for any exhaustion or only for those with controlled gradient length up to the boundary.
  3. In the applications to bounded symmetric domains, state explicitly which exhaustion function is used and verify that its gradient length satisfies the boundary hypothesis of the main result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central claim is a lower bound on the Kähler hyperbolicity modulus expressed in terms of the boundary behavior of |∇u| for a plurisubharmonic exhaustion function u. This is a conditional geometric estimate whose hypotheses (existence of suitable u on hyperconvex/strongly pseudoconvex domains) are part of the standard setup and do not reduce to a fitted parameter, self-definition, or self-citation chain. No equations or steps in the provided abstract reduce the claimed bound to its own inputs by construction. The derivation remains self-contained against external geometric properties and benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, new entities, or non-standard axioms are named. The result appears to rest on standard background facts of Kähler geometry and plurisubharmonic functions.

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discussion (0)

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Reference graph

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