arxiv: 2604.06437 · v1 · submitted 2026-04-07 · 🧮 math.CV · math.AP

Recognition: 2 theorem links

· Lean Theorem

Rarity of boldsymbol{mathcal{C}^{1,1}} solutions to the complex Monge--Amp\`ere equation on weakly pseudoconvex domains

Gautam Bharali, Rumpa Masanta

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification 🧮 math.CV math.AP

keywords complex Monge-Ampère equationweakly pseudoconvex domainsC^{1,1} regularityDirichlet problemplurisubharmonic functionsB-regular domainspotential theoryextension problems

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The pith

Smooth data for the complex Monge-Ampère Dirichlet problem fails to produce C^{1,1} solutions on weakly pseudoconvex domains.

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

The paper shows that the classical Dirichlet problem for the complex Monge-Ampère equation, when supplied with infinitely differentiable data, does not in general possess C^{1,1}-smooth solutions on weakly pseudoconvex B-regular domains. A reader would care because many problems involving holomorphic mappings and their extensions had been expected to reduce to the existence of such solutions. The result therefore directs attention toward potential-theoretic methods that avoid the need for this level of classical regularity.

Core claim

On any weakly pseudoconvex B-regular domain, the classical Dirichlet problem for the complex Monge-Ampère equation with C^∞-smooth data does not in general admit C^{1,1}-smooth solutions. This working draft serves as a prelude to potential-theoretic solutions for certain extension problems for mappings that were previously thought to rely on the existence of such C^{1,1}-smooth solutions.

What carries the argument

The Dirichlet problem for the complex Monge-Ampère equation, which asks for a plurisubharmonic function whose complex Hessian determinant equals a prescribed positive function and whose boundary values match given continuous data.

If this is right

Extension problems for mappings previously assumed to depend on C^{1,1} solutions must instead be treated by potential-theoretic techniques.
The expectation that C^∞ data automatically yields C^{1,1} regularity fails on these domains.
B-regular weakly pseudoconvex domains form a natural setting in which classical smoothness of solutions breaks down.

Where Pith is reading between the lines

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

Similar breakdowns in regularity may occur for other nonlinear PDEs on domains with weak boundary smoothness.
Potential theory could replace classical C^{1,1} solutions as the default tool for studying holomorphic extensions in several complex variables.
The result may indicate that C^{1,1} regularity is exceptional rather than generic for the complex Monge-Ampère equation outside strictly pseudoconvex domains.

Load-bearing premise

The domains must be weakly pseudoconvex and B-regular, which permits the existence of counterexamples where smooth data produces no C^{1,1} solution.

What would settle it

An explicit example of a weakly pseudoconvex B-regular domain together with smooth boundary data and a positive right-hand side for which the solution is provably C^{1,1} would contradict the claim that such solutions do not exist in general.

read the original abstract

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Amp\`ere equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such $\mathcal{C}^{1,1}$-smooth solutions.

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 shows that C^{1,1} solutions fail in general for the complex Monge-Ampère Dirichlet problem on any weakly pseudoconvex B-regular domain with smooth data, which is a useful negative result if the construction holds uniformly.

read the letter

The main point is that the authors prove a negative result: on every weakly pseudoconvex B-regular domain, there is smooth data for the complex Monge-Ampère equation whose solution is not C^{1,1}. They frame this as motivation to shift some extension problems toward potential-theoretic methods instead of relying on that regularity. That framing is straightforward and honest about the paper's purpose as a working draft. What looks new is the claim that the failure occurs for arbitrary domains in this class rather than for hand-picked examples. The abstract states the claim cleanly and ties it to downstream applications without overclaiming. The paper does well by being explicit that this is preliminary work meant to clear the way for other approaches. The soft spot is the one flagged in the stress-test note. To make the result hold for every such domain, the construction of the smooth right-hand side and the check that the Bedford-Taylor solution has unbounded complex Hessian must not depend on local coordinates or a defining function with smoothness beyond what weak pseudoconvexity and B-regularity give. If the argument slips in extra boundary regularity at any step, the universal quantifier does not go through. Without the details of the counterexample it is impossible to tell whether this is handled, but the concern is real and load-bearing. This paper is for people already working on regularity questions for the complex Monge-Ampère equation and on extension problems in several complex variables. A reader who needs to know the limitations of classical C^{1,1} assumptions will find it relevant. It deserves peer review because the claim, if it survives scrutiny on the construction, would affect how people set up these problems going forward.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that on any weakly pseudoconvex B-regular domain the classical Dirichlet problem for the complex Monge-Ampère equation with C^∞ data does not in general admit C^{1,1} solutions. It is presented as a working draft that motivates a shift to potential-theoretic methods for certain extension problems.

Significance. If established, the negative result would show that C^{1,1} regularity fails for some smooth data on every domain in this class, thereby limiting the applicability of classical solutions and supporting the paper's stated motivation for potential-theoretic alternatives. No machine-checked proofs or reproducible code are present.

major comments (1)

The universal quantifier ('any weakly pseudoconvex B-regular domain') requires an explicit, uniform construction of C^∞ data whose Bedford-Taylor solution has unbounded complex Hessian, without assuming extra boundary smoothness beyond the stated hypotheses. No such construction, local coordinate argument, or verification appears in the manuscript, rendering the central claim unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the identification of the key gap in our working draft. The manuscript is explicitly presented as a motivational prelude rather than a complete proof, and we will strengthen it accordingly.

read point-by-point responses

Referee: The universal quantifier ('any weakly pseudoconvex B-regular domain') requires an explicit, uniform construction of C^∞ data whose Bedford-Taylor solution has unbounded complex Hessian, without assuming extra boundary smoothness beyond the stated hypotheses. No such construction, local coordinate argument, or verification appears in the manuscript, rendering the central claim unsupported.

Authors: The referee is correct that the current draft states the universal claim without supplying the required explicit, uniform construction or verification. As a working draft whose primary purpose is to motivate a shift to potential-theoretic methods, the supporting details for the claim were left for later development. In the revised version we will add a uniform construction that applies to every weakly pseudoconvex B-regular domain, using only the stated hypotheses: B-regularity guarantees existence of the Bedford-Taylor solution, while the data are chosen (via a local argument near a boundary point of weak pseudoconvexity) so that the complex Hessian of the solution becomes unbounded. No extra boundary smoothness will be assumed. This directly remedies the gap identified by the referee. revision: yes

Circularity Check

0 steps flagged

No circularity: negative result on non-existence of C^{1,1} solutions stands independently of inputs

full rationale

The paper's central claim is a universal negative statement: for any weakly pseudoconvex B-regular domain, there exist C^∞ data for which the Dirichlet problem for the complex Monge-Ampère equation has no C^{1,1} solution. This is a direct non-existence result. No derivation chain is presented that reduces a 'prediction' or 'first-principles result' back to fitted parameters, self-definitions, or self-citations by construction. The abstract and setup invoke standard potential theory (Bedford-Taylor solutions) without redefining the solution class or smuggling ansatzes via prior self-work. The 'in general' qualifier is consistent with exhibiting counterexamples rather than fitting data. No load-bearing step equates the claimed rarity to an input by definition or renaming. The derivation is self-contained against external benchmarks in pluripotential theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are specified. The result relies on standard background notions in complex analysis such as weak pseudoconvexity, B-regularity, and the complex Monge-Ampère operator.

pith-pipeline@v0.9.0 · 5376 in / 1116 out tokens · 26405 ms · 2026-05-10T17:56:16.635610+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem 1.3 and 1.4: existence of φ ∈ C^∞(∂Ω) such that solution to (dd^c u)^n = f β^n is never C^{1,1} or C^{0,α} when D'Angelo type τ_1(ξ) ≥ 4 or weak Levi pseudoconvexity holds.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Use of Bedford-Taylor currents and maximum principle on psh functions; no cost functional or ratio symmetry appears.

Reference graph

Works this paper leans on

9 extracted references

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