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Weak solutions of the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases

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Pith's one-line read Weak solutions exist and are unique for the generalized Monge-Ampère and supercritical deformed Hermitian-Yang-Mills equations on the boundary of the solvable region.

desk verdict This paper closes the boundary cases for weak solutions of the generalized Monge-Ampère and supercritical dHYM equations using viscosity plus pluripotential methods. read the letter →

arxiv 2605.29258 v1 pith:4MCDHIUJ submitted 2026-05-28 math.DG

Rei Murakami This is my paper · ORCID
classification math.DG
keywords weaksolutionsMonge-AmpèreequationdeformedHermitian-Yang-Millsviscositymethodspluripotentialtheorygeometricflowscohomologyclassesboundarycases
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The pith

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

The reading

The paper proves existence and uniqueness of weak solutions to the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation for cohomology classes on the boundary of the solvable region. It also establishes that the associated geometric flows converge to these weak solutions in the sense of currents. The argument relies on combining viscosity-theoretic and pluripotential-theoretic techniques. A sympathetic reader would care because this completes the solvability picture by handling the previously excluded boundary cases in these nonlinear equations from complex geometry.

What carries the argument

The combination of viscosity-theoretic and pluripotential-theoretic techniques applied directly to boundary cohomology classes.

What would settle it

A specific cohomology class on the boundary of the solvable region for which no weak solution exists, or for which the associated geometric flow fails to converge in the sense of currents, would falsify the claim.

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Extended reading notes

Core claim

We prove the existence and uniqueness of weak solutions for the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover, we prove that the associated geometric flows converge to the weak solutions in the sense of currents. The proof combines viscosity-theoretic and pluripotential-theoretic techniques.

Load-bearing premise

The boundary of the solvable region in cohomology space remains a locus where viscosity and pluripotential techniques continue to apply without extra regularity or non-degeneracy assumptions.

Editorial extensions

If this is right

  • Weak solutions exist uniquely in boundary cohomology classes for both equations.
  • The geometric flows converge to these weak solutions in the sense of currents.
  • The combined techniques extend solvability results from the interior to the boundary without additional assumptions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The solvable region in cohomology space may be closed in the appropriate topology.
  • The boundary analysis could extend to related fully nonlinear equations on Kähler manifolds.
  • Convergence of flows at the boundary may allow study of limiting behavior exactly at the solvability threshold.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. The manuscript proves existence and uniqueness of weak solutions to the generalized Monge-Ampère equation and the supercritical deformed Hermitian-Yang-Mills equation for cohomology classes on the boundary of the solvable region. It further establishes convergence of the associated geometric flows to these weak solutions in the sense of currents, via a combination of viscosity-theoretic and pluripotential-theoretic techniques.

Significance. If the arguments hold, the result would complete the picture of solvability by handling the boundary locus, which is typically the most singular case. The dual use of viscosity and pluripotential methods is a methodological strength for treating weak solutions without interior non-degeneracy.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. We are pleased that the significance of completing the solvability picture for boundary classes and the dual use of viscosity and pluripotential methods are recognized as strengths. The recommendation is listed as uncertain, but no specific major comments are provided in the report. We therefore have no point-by-point responses to offer at this stage and would welcome any concrete concerns the referee may have so that we can address them directly.

Circularity Check

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No significant circularity detected

full rationale

The abstract states a proof of existence and uniqueness of weak solutions for the generalized Monge-Ampère and supercritical deformed Hermitian-Yang-Mills equations on the boundary of the solvable region, using a combination of viscosity-theoretic and pluripotential-theoretic techniques, plus convergence of associated flows in the sense of currents. No equations, parameter fits, self-citations, ansatzes, or derivation steps are supplied in the given material. The central claim is an extension of standard techniques to a boundary case without any visible reduction of outputs to inputs by construction, self-definition, or load-bearing self-citation. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

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Cite this review

Pith. "Pith review of Weak solutions of the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases." pith.science (2026). https://pith.science/paper/4MCDHIUJ

@misc{pith2026260529258,
  author       = {Pith},
  title        = {Pith review of: Weak solutions of the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4MCDHIUJ}},
  note         = {Machine review of arXiv:2605.29258}
}
read the original abstract

We prove the existence and uniqueness of weak solutions for the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover, we prove that the associated geometric flows converge to the weak solutions in the sense of currents. The proof combines viscosity-theoretic and pluripotential-theoretic techniques.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Datar-Mete-Song minimal slope conjecture

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    The paper proves the Datar-Mete-Song conjecture: a pair of Kähler classes is semi-stable exactly when its minimal J-slope equals the topological J-slope.

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