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arxiv: 2604.13421 · v1 · submitted 2026-04-15 · 🧮 math.CV · math.AP· math.DG

Some variational problems for the complex Monge--Amp{\`e}re operator

Jianchun Chu , Yaxiong Liu , Nicholas McCleerey , Weijun Zhang This is my paper

Pith reviewed 2026-05-10 12:33 UTC · model grok-4.3

classification 🧮 math.CV math.APmath.DG
keywords complex Monge-Ampère equationDirichlet problemstrongly pseudoconvex manifoldsKähler manifoldsparabolic flowexistence of smooth solutionsvariational problems
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The pith

Smooth solutions exist for the complex Monge-Ampère Dirichlet problem on strongly pseudoconvex Kähler manifolds when the right-hand side decreases in the solution.

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

The paper addresses the Dirichlet problem for the complex Monge-Ampère equation on strongly pseudoconvex Kähler manifolds where the right-hand side decreases as a function of the unknown. Flow-based arguments are used to prove existence of smooth solutions in several natural cases, extending earlier work of Chou and Wang. This matters to readers interested in complex geometry because the equation controls the existence of Kähler metrics with prescribed volume forms or curvature properties. The monotonicity assumption on the right-hand side allows the flow to converge without additional barriers.

Core claim

Using flow-based arguments, we establish existence of smooth solutions to the Dirichlet problem for the complex Monge-Ampère equation on strongly pseudoconvex Kähler manifolds when the right-hand side is decreasing in the solution.

What carries the argument

Parabolic flow method that deforms an initial function to a smooth solution of the elliptic complex Monge-Ampère equation.

If this is right

  • The Dirichlet problem admits smooth solutions whenever the right-hand side decreases in the unknown and the manifold satisfies the geometric hypotheses.
  • The parabolic flow converges to a solution, giving a constructive approximation procedure.
  • The result applies to multiple natural choices of decreasing right-hand sides that arise in geometric problems.

Where Pith is reading between the lines

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

  • The flow technique could be tested numerically on simple domains to check convergence rates for specific decreasing nonlinearities.
  • Similar monotonicity conditions might permit existence results for related fully nonlinear equations on Kähler manifolds.
  • The existence statements could be used to construct Kähler metrics whose volume forms satisfy decreasing dependence on the potential.

Load-bearing premise

The manifold must be strongly pseudoconvex and Kähler, and the right-hand side must be decreasing in the solution.

What would settle it

A concrete strongly pseudoconvex Kähler manifold together with a smooth decreasing right-hand side for which the Dirichlet problem has no smooth solution would disprove the claim.

read the original abstract

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth solutions in a number of natural circumstances, following work of Chou-Wang.

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 / 1 minor

Summary. The manuscript considers the Dirichlet problem for the complex Monge-Ampère equation on strongly pseudoconvex Kähler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, the authors establish existence of smooth solutions in a number of natural circumstances, following the work of Chou and Wang.

Significance. If the a priori estimates, boundary regularity, and convergence arguments hold, this work usefully extends the Chou-Wang parabolic flow framework by exploiting the monotonicity condition to obtain a comparison principle and convergence to a stationary solution. The result strengthens existence theory for the complex Monge-Ampère equation under natural structural assumptions and may apply to related problems in Kähler geometry.

minor comments (1)
  1. [Introduction] The title refers to 'variational problems' while the abstract and central result focus on the Dirichlet problem; a short paragraph or sentence in the introduction linking the flow method to an underlying variational structure would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point at this stage. We will carefully review the manuscript for any minor issues such as typos or clarifications before resubmission.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes existence of smooth solutions to the complex Monge-Ampère Dirichlet problem via parabolic flow methods on strongly pseudoconvex Kähler manifolds under the monotonicity assumption on the right-hand side. This extends the Chou-Wang framework with independent a priori estimates and boundary regularity arguments. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain; the comparison principle and convergence follow from the stated PDE assumptions without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of strongly pseudoconvex Kähler manifolds and the monotonicity assumption on the right-hand side; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The manifold is strongly pseudoconvex and Kähler
    Explicitly stated as the geometric setting for the Dirichlet problem.
  • domain assumption The right-hand side is decreasing in the solution
    The key structural hypothesis that enables the flow argument.

pith-pipeline@v0.9.0 · 5348 in / 1147 out tokens · 35660 ms · 2026-05-10T12:33:09.601176+00:00 · methodology

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Reference graph

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