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arxiv: 2606.18740 · v1 · pith:PG3FN5A6new · submitted 2026-06-17 · 🧮 math.DG · math.CV

Bottom of the spectrum of complete noncompact K\"{a}hler manifolds

Ye-Won Luke Cho , Young-Jun Choi This is my paper

Pith reviewed 2026-06-26 20:10 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords bottom of the spectrumHodge LaplacianKähler manifoldKähler hyperbolicbounded symmetric domainrigiditycurvature boundnoncompact manifold
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The pith

Complete noncompact Kähler manifolds have their Hodge Laplacian spectrum bottom bounded above by curvature conditions, with rigidity at the maximum.

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

This survey collects theorems on the bottom of the spectrum of the Hodge Laplacian for complete noncompact Kähler manifolds. Emphasis falls on Kähler hyperbolic manifolds and bounded symmetric domains, where explicit control is possible. The results include upper bounds derived from Ricci curvature and holomorphic bisectional curvature assumptions, together with rigidity statements that characterize manifolds attaining the largest possible value. Several open problems are stated as directions for further study.

Core claim

The paper presents theorems establishing upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions on complete noncompact Kähler manifolds. For the special classes of Kähler hyperbolic manifolds and bounded symmetric domains these bounds are attained, and rigidity results identify the manifolds that achieve the maximal value. The survey organizes these statements and lists open problems that remain after the known results.

What carries the argument

The bottom of the spectrum of the Hodge Laplacian, which the survey controls via curvature assumptions and shows to be rigid at its upper limit for hyperbolic and symmetric cases.

If this is right

  • Negative Ricci curvature implies an explicit upper bound on the bottom of the spectrum.
  • Negative holomorphic bisectional curvature likewise yields an upper bound on the same quantity.
  • Equality in either bound forces the manifold to be rigid, typically isometric to a model space in the Kähler hyperbolic or bounded symmetric domain classes.
  • The same rigidity statements apply when the manifold belongs to the emphasized special classes.

Where Pith is reading between the lines

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

  • The pattern of curvature-controlled spectral bounds may suggest similar controls for the spectrum of the Dirac operator on the same manifolds.
  • Numerical verification of the bounds on explicit examples such as quotients of the complex hyperbolic plane could test the sharpness statements.
  • The open problems listed may link to questions about the spectrum on non-Kähler Hermitian manifolds with analogous curvature conditions.

Load-bearing premise

The manifolds are complete noncompact Kähler manifolds that satisfy the stated curvature bounds or hyperbolicity conditions.

What would settle it

A complete noncompact Kähler manifold obeying negative Ricci or bisectional curvature whose bottom of the spectrum lies strictly above the upper bound given by the theorems would contradict the surveyed results.

read the original abstract

We present a survey on the bottom of the spectrum of the Hodge Laplacian on complete noncompact K\"ahler manifolds, with particular emphasis on K\"ahler hyperbolic manifolds and bounded symmetric domains. We also discuss theorems regarding the upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions, along with rigidity results for manifolds attaining the maximal bottom of the spectrum. Throughout the article, we propose several open problems.

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. This manuscript is a survey on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds. It emphasizes Kähler-hyperbolic manifolds and bounded symmetric domains, presents theorems regarding upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions, discusses rigidity results for manifolds attaining the maximal bottom of the spectrum, and proposes several open problems.

Significance. If the compilation accurately and comprehensively restates the cited results from the literature, the survey would serve as a useful reference for researchers in Kähler geometry by organizing known theorems on spectral properties under curvature conditions and by identifying open questions that could guide future work.

minor comments (3)
  1. The abstract and introduction should explicitly note that the theorems discussed are restatements of existing results from the literature rather than new contributions by the authors.
  2. A dedicated section or subsection listing the proposed open problems would improve readability and highlight their role in the survey.
  3. Ensure that all theorem statements include precise citations to the original sources to facilitate verification by readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our survey and the recommendation of minor revision. The report does not list any specific major comments requiring point-by-point responses.

Circularity Check

0 steps flagged

Survey paper with no original derivations or predictions

full rationale

The manuscript is explicitly a literature survey presenting theorems from prior work on the bottom of the spectrum for complete noncompact Kähler manifolds, with emphasis on Kähler-hyperbolic cases and bounded symmetric domains. It discusses existing upper bounds under Ricci/bisectional curvature assumptions and associated rigidity results, while proposing open problems. No new theorems, derivations, predictions, or first-principles results are advanced by the authors, so no derivation chain exists that could reduce to inputs by construction. The central content is accurate restatement of external literature, rendering the paper self-contained against external benchmarks with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a survey and introduces no new free parameters, axioms, or invented entities beyond those already present in the referenced literature on Kähler geometry.

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discussion (0)

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Reference graph

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