A new approach to the Monge-Amp\`ere eigenvalue problem

Ahmed Zeriahi; Chinh H. Lu

REVIEW 1 major objections 5 minor 43 references

A new approach to the Monge-Amp\`ere eigenvalue problem

T0 review · 1 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For any non-pluripolar measure, a Monge-Ampère eigenvalue pair is unique up to a positive constant.

desk verdict A real but repairable gap in Theorem 3.8; the envelope method is genuinely new and the paper deserves peer review after a major revision. read the letter →

arxiv 2507.18409 v1 pith:5ZA7VMA6 submitted 2025-07-24 math.CV math.AP

Chinh H. Lu , Ahmed Zeriahi This is my paper · ORCID

classification math.CVmath.AP MSC 31C4532U1532U4032W2035J6635J96

keywords complexMonge-AmpereoperatorDirichleteigenvalueuniquenessiterativemethodplurisubharmonicenvelopeCegrellfiniteenergyclassnon-pluripolarmeasurereal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that, whenever a solution exists, the complex Monge-Ampère eigenvalue problem in a bounded hyperconvex domain is essentially unique: a single eigenvalue $\lambda_1(\mu)$ and eigenfunctions that are all positive multiples of one another, for any non-pluripolar positive Borel measure $\mu$. The eigenvalue is identified as a Rayleigh quotient, $\lambda_1(\mu)=\inf E(u)/I_\mu(u)$ over the Cegrell finite-energy class of negative plurisubharmonic functions with finite Monge-Ampère energy, with no regularity assumptions on $\mu$. A new envelope construction partially linearizes the nonlinear equation, replacing the smooth-solution arguments used in earlier work. Under a natural continuity assumption on $\mu$, the paper also shows that an iterative scheme started from any negative finite-energy function converges to the eigenvalue and eigenfunction. The method extends to complex Hessian operators and, via a logarithmic transformation, to the real Monge-Ampère operator.

What carries the argument

The load-bearing mechanism is the plurisubharmonic envelope $P(h)=(\sup\{v\in\mathrm{PSH}(\Omega): v\le h\})^*$, applied to $\min(u-\psi,0)$ to combine a solution and a subsolution into a supersolution whose Monge-Ampère measure is concentrated on the contact set $\{v=u-\psi\}$. Lemma 3.4 then uses the variational definition of $\lambda_1$ to show that any supersolution at the exact quotient $\lambda_1$ is a true solution: integrating the supersolution inequality against $(-u)$ forces equality because $E(u)\le\lambda_1^n I_\mu(u)$ and $\lambda_1\le E/I_\mu$. This partial sublinearization is the step that lets the argument run without a smooth solution to linearize around.

What would settle it

Assume there is a $\psi$ satisfying $(dd^c\psi)^n\ge(-\lambda_1\psi)^n\mu$ strictly on a set of positive $\mu$-measure, construct $u$ by solving $(dd^c u)^n=(-\lambda_1\psi)^n\mu$, and test whether $(dd^c u)^n\le(-\lambda_1 u)^n\mu$ holds on the contact set $\{u=\psi\}$; a single such measure would show the general-case uniqueness assertion needs an additional hypothesis.

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Extended reading notes

Core claim

The paper's central theorem states that if $(\lambda,\varphi)$ solves $(dd^c u)^n=(-\lambda u)^n\mu$ with $u$ in the Cegrell finite-energy class $E^1(\Omega)$, then $\lambda=\lambda_1(\mu)=\inf\{E(u)/I_\mu(u): u\in E^1(\Omega)\setminus\{0\}\}$, and any $\psi\in E^1(\Omega)\setminus\{0\}$ satisfying $(dd^c\psi)^n\ge(-\lambda\psi)^n\mu$ is in fact a solution and equals $c\varphi$ for some $c>0$. This removes the smoothness assumptions on the domain and the density that earlier uniqueness results required; $\mu$ only has to vanish on pluripolar sets. In particular the inequality cannot be strict: a function that merely lies on the subsolution side at the critical quotient is already an eigenfunction. When $I_\mu$ is continuous on the energy sublevel sets, the same framework proves existence and gives an iterative approximation of the eigenpair.

Load-bearing premise

The proof of the general uniqueness statement in the general-case part of Theorem 3.8 rests on the premise that the auxiliary function $u$ solving $(dd^c u)^n=(-\lambda\psi)^n\mu$ with $u\ge\psi$ is a supersolution at $\lambda_1$; Lemma 3.4 can convert a supersolution into a solution, but not a mere subsolution, so if that inequality is missing the uniqueness conclusion for arbitrary non-pluripolar measures is not established.

Editorial extensions

If this is right

The first eigenvalue $\lambda_1(\mu)$ is a genuine invariant of the pair $(\Omega,\mu)$: no second eigenvalue can exist, and all eigenfunctions are proportional.
Lions' original strategy is valid in this generality: $\lambda_1$ is the largest $\lambda$ for which $(dd^c u)^n=(1-\lambda u)^n\mu$ has a finite-energy solution.
When $I_\mu$ is continuous on each energy sublevel set, the iteration $(dd^c u_{k+1})^n=R(u_k)(-u_k)^n\mu$ with $R(u_k)=E(u_k)/I_\mu(u_k)$ converges from any nonzero finite-energy start to a solution of the eigenvalue problem.
The same uniqueness, variational formula, and iterative convergence hold for the complex Hessian operators $H_m(u)=(dd^c u)^m\wedge\beta^{n-m}$ on $m$-hyperconvex domains.
For the real Monge-Ampère operator on a bounded convex domain, the eigenvalue pair is unique with no regularity hypothesis, and existence holds for measures that are the real Monge-Ampère measure of a continuous convex function, and more generally for any measure integrating a negative convex function.

Reading between the lines

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

Because the proof only needs envelopes and the variational definition, the same partial-sublinearization recipe may apply to other fully nonlinear Dirichlet eigenvalue problems that admit a comparison principle.
The quotient $\lambda_1(\mu)$ behaves like a spectral radius for a nonlinear operator, so one could ask whether eigenfunction level sets encode the geometry of the contact set when $\mu$ is purely singular.
The monotonicity of $E(u_k)$ along the iteration suggests a stable numerical method for measures without densities; testing it on singular measures could show how close the continuity assumption on $I_\mu$ is to being necessary.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The reader's central complaint is real but smaller than it first looks. There is a genuine gap in the proof of Theorem 3.8: the function u constructed in the general case is only shown to be a subsolution, and Lemma 3.4 only upgrades supersolutions. That is a load-bearing error in the written proof. But it is repairable with material already in the paper. Since (dd^c u)^n = (−λψ)^n μ is absolutely continuous with respect to μ, the earlier absolutely continuous case of Theorem 3.8 applies directly to u, giving u a solution and φ proportional to u. Equality of measures then forces ψ = u, so ψ solves. The repair is a missing paragraph, not a missing idea.

What is actually new is worth taking seriously. The envelope method yields the Rayleigh quotient formula and uniqueness of eigenfunctions for arbitrary non-pluripolar measures, with no smoothness assumptions on the data. The iterative scheme is genuinely monotone and does not presuppose existence of an eigenfunction. The real Monge-Ampère results via the logarithmic map are a nice translation, and the Hessian extension is natural. The variational formula in Theorem 3.5 and the absolutely continuous case of Theorem 3.8 are solid. The citations to GLZ19, Zer25, BZ23 are to published work with independent derivations; no circularity.

The soft spots are proportionate. The Hessian theorem (6.1) is asserted with the proof left to the reader; for a paper advertising a new method, that is too terse. The same applies to the Hessian analogues of Theorem 4.4. The real-domain section inherits the gap, so it will need the same repair. There are also small typos (e.g. 'pluribubharmonic' in Section 2.1) and a possible misreference in Proposition 7.3's proof. None of these are fatal.

Who is this for? Anyone working on nonlinear eigenvalue problems, pluripotential theory, or the variational theory of Monge-Ampère equations. It deserves a serious referee: the results are important, the method is fresh, and the gap is localized and fixable. I would send it to review with the expectation of a major revision, not desk reject. If the authors add the missing argument in Theorem 3.8 and at least sketch the Hessian proof, the paper will be a solid contribution.

Referee Report

1 major / 5 minor

Summary. The paper studies the complex Monge-Ampère eigenvalue problem in bounded hyperconvex domains for a non-pluripolar positive Borel measure μ. The central results are: a Rayleigh-quotient formula for the eigenvalue λ1(μ) in terms of the Cegrell energy E and the functional Iμ(u)=∫(-u)^{n+1}dμ; uniqueness of eigenfunctions in E1(Ω) up to positive constants with no regularity assumption on μ; an existence and approximation result via an iterative scheme under a continuity assumption on Iμ; a general Dirichlet problem with non-monotone right-hand side; and analogues for complex Hessian and real Monge-Ampère operators. The method is based on plurisubharmonic envelopes and a comparison/supersolution argument rather than on linearization around a smooth solution.

Significance. If the main theorem is correct, the paper gives a substantial generalization of earlier smooth-data results of Lions and of Badiane–Zeriahi: uniqueness of eigenfunctions and a variational formula are obtained for arbitrary non-pluripolar measures, with no boundary or density assumptions. The envelope method is a genuinely new tool in this problem, and the extensions to Hessian and real operators are natural and potentially useful. The variational formula is derived rather than assumed, so there is no circularity in the definition of the eigenvalue. The absolutely continuous case of the uniqueness proof appears sound, as does the Rayleigh quotient argument. However, as detailed below, the general non-absolutely-continuous case of the uniqueness proof contains a load-bearing gap that, as written, invalidates Theorem 1.1's full uniqueness claim and the subsequent corollaries.

major comments (1)

[§3.2, proof of Theorem 3.8, general case] The step 'It thus follows from Lemma 3.4 that (λ,u) solves (MA_{μ,λ})' is not justified. The constructed function u satisfies (dd^c u)^n = (−λψ)^n μ. Since u≥ψ and both are negative, (−λψ)^n μ ≥ (−λu)^n μ, so u is a subsolution of (MA_{μ,λ}), not a supersolution. Lemma 3.4, the only tool invoked to upgrade u to a solution, explicitly requires the supersolution inequality (dd^c u)^n ≤ (−λu)^n μ. Consequently the identity E(u)/Iμ(u)=λ^n, which is used to obtain the contradiction with the positive measure σ, is not established. This is the precise step that removes the absolute-continuity assumption on (dd^c ψ)^n, so Theorem 1.1, Corollary 3.9, and the real-variable Theorem 7.5 are not proved as written. The gap appears localized and repairable: since (dd^c u)^n = (−λψ)^n μ ≪ μ, the first, absolutely continuous case of Theorem 3.8 can be applied to the subsolution u, yielding that u is a solution and u=cφ; one then still needs to close the argument to identify ψ with a multiple of φ. The authors should supply the missing reasoning carefully.

minor comments (5)

[§2.1] There is a typo: 'pluribubharmonic' should be 'plurisubharmonic'.
[§2.2] The phrase 'in the sens of Borel measures' appears twice; 'sens' should be 'sense'.
[§3.1, proof of Theorem 3.5] In the line 'E(u)−γ1^n Iμ(u)=0≤E(v)−μ1 Iμ(v)', the symbol 'μ1' should presumably be 'γ1'.
[§4.1, proof of Theorem 4.1] The word 'leaded' in 'which leaded to (4.2)' should be 'led'.
[§6] Theorem 6.1 is an advertised extension to complex Hessian operators, but its proof is only a sentence saying that the entire proof can be adapted and 'details are left to the reader'. For a refereed journal, the adaptation should be written out, at least in outline, so that the reader can verify that the envelope lemmas and the comparison arguments carry over without new hypotheses.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Rayleigh quotient formula is a definition and is proved to coincide with any eigenvalue; the cited prior results are independent published theorems, and the questionable step in Theorem 3.8 is a derivation gap rather than a circular reduction.

full rationale

The paper's central claim is not circular. The quantity λ1(μ) is introduced in Definition 3.1 as the infimum of the Rayleigh quotient E(u)/Iμ(u), and Theorem 3.5 proves separately, via envelope and domination arguments, that any solution of the eigenvalue problem has λ = λ1(μ). The eigenfunction uniqueness result in Theorem 3.8 is then proved from that variational identity and from comparison/envelope lemmas; it does not define the eigenvalue in terms of the eigenfunction. The dependence on prior work by the same authors ([GLZ19], [Zer25], [BZ23], [ÅCLR24]) is real but does not amount to circularity: those results are published, parameter-free theorems with stated assumptions that do not include the present eigenvalue claim, and they are used as tools rather than as the conclusion being derived. The one logically problematic passage is in the general case of Theorem 3.8, where the authors construct u with (ddcu)^n = (−λψ)^n μ, observe u is a subsolution, and then write 'It thus follows from Lemma 3.4 that (λ,u) solves (MAμ,λ)'; Lemma 3.4 converts supersolutions into solutions, not subsolutions. This is a genuine gap in the written proof, but it is a correctness or completeness issue, not an instance of circularity: the missing inequality (ddcu)^n ≤ (−λu)^n μ is not supplied by any definitional identity or fitted input, and the gap appears repairable by applying the previously proved absolutely continuous case to u. Under the hard rules, derivation gaps without an equation-to-equation or input-to-output identity do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests on standard pluripotential theory and several heavy external results (envelope theorem, ACC existence, real MA Dirichlet solvability). No free parameters are fitted to data, and no new entities are invented. The main assumptions are the non-pluripolarity of μ and, for the iterative part, the continuity of Iμ on energy sublevel sets.

assumptions (5)

standard math The Bedford-Taylor theory of the complex Monge-Ampère operator on bounded hyperconvex domains and the Cegrell class E^1(Ω).
Used throughout Sections 2 and 3 to define the operator, energy functional, and the variational space.
standard math The plurisubharmonic envelope theorem for quasi-continuous functions (Theorem 2.8), cited from [GLZ19].
This is the engine of the new method; it guarantees the envelope is in E^1 and its Monge-Ampère measure is carried by the contact set.
standard math Existence of solutions to (dd^c u)^n = ν for measures ν satisfying the a priori estimate in [˚ACC12, Theorem A].
Used in the general case of Theorem 3.8 to construct the auxiliary function u.
standard math Known solvability of the real Monge-Ampère equation MR(u)=ν with ν(D)<∞ (Theorem 7.2, cited to [Har06] and others).
Used in Section 7 to build approximate solutions on compact subsets and to convert the complex solution back to the real domain.
standard math The comparison principle and domination principle for the complex Monge-Ampère operator in E^1.
Quoted as Propositions 2.4, 2.5, and 2.6 and used in many steps.

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Pith. "Pith review of A new approach to the Monge-Amp\`ere eigenvalue problem." pith.science (2026). https://pith.science/paper/5ZA7VMA6

@misc{pith2026250718409,
  author       = {Pith},
  title        = {Pith review of: A new approach to the Monge-Amp\`ere eigenvalue problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZA7VMA6}},
  note         = {Machine review of arXiv:2507.18409}
}

abstract

We study the eigenvalue problem for the complex Monge-Amp\`ere operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the finite energy class introduced by Cegrell, up to positive multiplicative constants, and provide a Rayleigh quotient type formula for computing the eigenvalue. Under a natural continuity assumption on the measure, we further show that both the eigenvalue and eigenfunctions can be obtained via an iterative procedure starting from any negative finite energy function. Our approach relies on the fine properties of plurisubharmonic envelopes, which allow a partial sublinearization of the nonlinear problem. As far as we know, this method is new, even in the linear case, and not only yields new results but also significantly simplifies existing arguments in the literature. Moreover, it extends naturally to the setting of complex Hessian operators. Finally, by translating our results from the complex Monge-Amp\`ere setting via a logarithmic transformation, we also obtain several interesting analogues for the real Monge-Amp\`ere operator.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

On Dirichlet 's principle and problem

Per hag, Urban Cegrell, and Rafa Czy \.z . On Dirichlet 's principle and problem. Math. Scand. , 110(2):235--250, 2012

work page 2012

[2] [2]

Lu, and Alexander Rashkovskii

Per hag, Rafa Czy \.z , Chinh H. Lu, and Alexander Rashkovskii. Kiselman minimum principle and rooftop envelopes in complex Hessian equations. Math. Z. , 308(4):24, 2024. Id/No 70

work page 2024

[3] [3]

Lu, and Alexander Rashkovskii

Per hag, Rafa Czy \.z , Chinh H. Lu, and Alexander Rashkovskii. Geodesic connectivity and rooftop envelopes in the Cegrell classes. Math. Ann. , 391(3):3333--3361, 2025

work page 2025

[4] [4]

Inverse iteration for the Monge - Amp \`e re eigenvalue problem

Farhan Abedin and Jun Kitagawa. Inverse iteration for the Monge - Amp \`e re eigenvalue problem. Proc. Am. Math. Soc. , 148(11):4875--4886, 2020

work page 2020

[5] [5]

A. D. Alexandrov and V. A. Zalgaller. Intrinsic Geometry of Surfaces. Translations of Mathematical Monographs, Amer. Math. Soc , 15, 1967

work page 1967

[6] [6]

Berman and Bo Berndtsson

Robert J. Berman and Bo Berndtsson. Real Monge - Amp \`e re equations and K \"a hler - Ricci solitons on toric log Fano varieties. Ann. Fac. Sci. Toulouse, Math. (6) , 22(4):649--711, 2013

work page 2013

[7] [7]

Berman, S \'e bastien Boucksom, Vincent Guedj, and Ahmed Zeriahi

Robert J. Berman, S \'e bastien Boucksom, Vincent Guedj, and Ahmed Zeriahi. A variational approach to complex Monge - Amp \`e re equations. Publ. Math., Inst. Hautes \'E tud. Sci. , 117:179--245, 2013

work page 2013

[8] [8]

Plurisubharmonic functions with weak singularities

Slimane Benelkourchi, Vincent Guedj, and Ahmed Zeriahi. Plurisubharmonic functions with weak singularities. In Complex analysis and digital geometry. Proceedings from the Kiselmanfest, Uppsala, Sweden, May 2006 on the occasion of Christer Kiselman's retirement , pages 57--74. Uppsala: Univ. Uppsala, 2009

work page 2006