{"id":"f656d237-e7f7-494e-b9f1-fd88c19687d9","arxiv_id":"2507.18409","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-pluripolar measures on hyperconvex domains, the complex Monge-Ampère eigenvalue is unique, eigenfunctions are proportional, and the eigenvalue is the infimum of a Rayleigh quotient.","lead":"The paper proves uniqueness and a Rayleigh quotient formula for the complex Monge-Ampère eigenvalue problem under very weak assumptions on the measure. It introduces a plurisubharmonic envelope method that also yields an iterative algorithm and transfers results to the real Monge-Ampère operator.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8's general case invokes Lemma 3.4 for a function that is only a subsolution, so the uniqueness proof is incomplete as written; however the earlier absolutely continuous case appears to repair the gap.","rationale":"The reader's weakest assumption identifies exactly the invalid step in the general case of Theorem 3.8: Lemma 3.4 is applied to a function u for which only the subsolution inequality has been shown. I agree that this breaks the proof as written and that the Rayleigh quotient formula and the absolutely continuous case are not the source of the problem. I disagree only on the severity of the consequence. The gap is not a counterexample to the theorem; the paper's own first case, applied to u, provides the missing upgrade to a solution, after which the existing contradiction with σ closes. Thus the central claim is likely correct, but the submitted proof is incomplete. A conditional acceptance requiring the authors to supply the one-paragraph repair is more accurate than an outright reject. Secondary issues, such as Theorem 6.1 being asserted with details left to the reader, reinforce the need for revision but are not the primary obstacle.","tokens_in":28967,"tokens_out":24314,"duration_ms":248520,"concrete_test":"Rewrite the final paragraph of the proof of Theorem 3.8 without citing Lemma 3.4. After constructing u, apply the already-proved absolutely continuous case of Theorem 3.8 to the subsolution u (valid since (dd^c u)^n ≪ μ); this should give (dd^c u)^n = (−λu)^n μ and φ=cu. Then check the existing contradiction: λ_1(μ+σ)=λ and E(u)/I_μ(u)=λ^n yield λ^n > E(u)/I_{μ+σ}(u) ≥ (λ_1(μ+σ))^n = λ^n. If this replacement works, the central theorem stands and only the proof needs revision; if it cannot be made to run, the uniqueness theorem is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the general case of Theorem 3.8, the paper constructs u∈E^1(Ω) with (dd^c u)^n = (−λψ)^n μ. Since ψ is a subsolution and u≥ψ, the construction gives (dd^c u)^n = (−λψ)^n μ ≥ (−λu)^n μ, i.e. u is a subsolution in the sense of §3. Lemma 3.4, the only tool cited to upgrade u to a solution, requires the opposite inequality (supersolution), and no proof of that inequality is supplied. Consequently the equality E(u)/I_μ(u)=λ^n, on which the contradiction with the positive measure σ rests, is not established; the paper never shows that the auxiliary function u solves (MA_{μ,λ}). Since this is the step that handles subsolutions whose Monge-Ampère measure is not absolutely continuous with respect to μ, Theorem 1.1's uniqueness claim is not proved as written. The gap is localized and appears repairable: because (dd^c u)^n = (−λψ)^n μ ≪ μ, the first (absolutely continuous) case of Theorem 3.8 can be applied to u, yielding that u solves and φ=cu; the subsequent argument with μ+σ then closes the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29152,"tokens_out":9789,"duration_ms":104463,"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":[{"comment":"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.","section":"§3.2, proof of Theorem 3.8, general case"}],"minor_comments":[{"comment":"There is a typo: 'pluribubharmonic' should be 'plurisubharmonic'.","section":"§2.1"},{"comment":"The phrase 'in the sens of Borel measures' appears twice; 'sens' should be 'sense'.","section":"§2.2"},{"comment":"In the line 'E(u)−γ1^n Iμ(u)=0≤E(v)−μ1 Iμ(v)', the symbol 'μ1' should presumably be 'γ1'.","section":"§3.1, proof of Theorem 3.5"},{"comment":"The word 'leaded' in 'which leaded to (4.2)' should be 'led'.","section":"§4.1, proof of Theorem 4.1"},{"comment":"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.","section":"§6"}],"recommendation":"major_revision","confidential_remarks":"The main gap in the proof of Theorem 3.8 is localized and appears repairable, so I recommend major revision rather than rejection. The refereeing note that the earlier absolutely continuous case is sound and can likely be applied to the auxiliary function u is convincing. I would ask the authors to fix the general-case proof of Theorem 3.8 and to expand the proof of the Hessian theorem in Section 6. The paper relies heavily on prior work by the same authors and collaborators [Zer25], [GLZ19], and [BZ23], but those are published and properly cited, so I do not see a circularity or prior-knowledge problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe 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.\n\nWhat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":29752,"tokens_out":6538,"would_cite":true,"duration_ms":62373,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["31C45","32U15","32U40","32W20","35J66","35J96"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any non-pluripolar measure, a Monge-Ampère eigenvalue pair is unique up to a positive constant.","keywords":["complex Monge-Ampere operator","Dirichlet eigenvalue","uniqueness","iterative method","plurisubharmonic envelope","Cegrell finite energy class","non-pluripolar measure","real Monge-Ampere operator"],"falsifier":"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.","tokens_in":28684,"feed_emoji":"📐","tokens_out":14909,"duration_ms":135229,"temperature":0.7,"pith_summary":"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.","feed_headline":"Monge-Ampère eigenfunctions are unique for any non-pluripolar measure","feed_subtitle":"A Rayleigh-quotient formula computes the eigenvalue with no smoothness assumptions on the measure.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduced the real Monge-Ampère eigenvalue problem and the strategy of solving $(dd^c u)^n=(1-\\lambda u)^n\\mu$ that the paper extends.","marker":"[Lio85]"},{"why":"Established uniqueness of eigenfunctions under smooth data in the complex case, the result here generalized to arbitrary non-pluripolar measures.","marker":"[BZ23]"},{"why":"Supplies the envelope theorem stating that $P(h)$ lies in $E^1$ and its Monge-Ampère measure is concentrated on the contact set, the key input to Lemma 3.3.","marker":"[GLZ19]"},{"why":"Provides the Cegrell energy classes, the comparison principle, and the solvability criteria used throughout the paper.","marker":"[Ceg98]"},{"why":"Provides Theorem A, used in the general case of Theorem 3.8 to construct $u$ solving $(dd^c u)^n=(-\\lambda\\psi)^n\\mu$.","marker":"[˚ACC12]"},{"why":"Showed the iterative scheme converges when an eigenfunction is known to exist; this paper removes that prior existence assumption.","marker":"[Zer25]"},{"why":"Proved the Rayleigh-quotient formula in the real setting that the paper adapts to the complex case.","marker":"[Tso90]"}],"fun_headline_variants":["New envelope proof simplifies Monge-Ampère eigenproblem","Rayleigh quotient formula for Monge-Ampère eigenvalues","Iterative method converges to Monge-Ampère eigenpairs","No smoothness needed: unique Monge-Ampère eigenfunctions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New envelope proof simplifies Monge-Ampère eigenproblem","Rayleigh quotient formula for Monge-Ampère eigenvalues","Iterative method converges to Monge-Ampère eigenpairs","No smoothness needed: unique Monge-Ampère eigenfunctions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00134,"raw_usage":{"total_tokens":5456,"prompt_tokens":962,"completion_tokens":4494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":4423}},"tokens_in":578,"tokens_out":4494,"duration_ms":33701,"temperature":1.0,"reasoning_tokens":4423,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:14:16.304112+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}