{"id":"cd2a51e3-eaaa-473b-be51-dcd5194880ed","arxiv_id":"2507.13273","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An inverse-iteration sequence converges uniformly to the eigenfunction of the complex Monge-Ampere Dirichlet eigenvalue problem, with Rayleigh quotients decreasing to the first eigenvalue.","lead":"This paper constructs an iterative method that, starting from a simple initial function, approximates the first eigenvalue and eigenfunction of the complex Monge-Ampere equation on a bounded strictly pseudoconvex domain. The result extends a known real-variable technique to the complex setting and gives a constructive proof that the iterations converge uniformly to the solution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the uniform Hölder step is the natural focal point, but the L^p bound and the cited regularity theorems close it.","rationale":"The reader identified the uniform Hölder estimates (3.6) as the weakest assumption, and that is indeed the step to scrutinize: if the L^{1+1/n} norms of f_k were not uniformly bounded, the Arzelà-Ascoli compactness argument would collapse. In my reading, however, the bound is valid. The computation in §3.5 has a typo in the exponent of ||f||∞, but the corrected exponent 1/n is also finite and the monotonicity lemma gives a uniform bound on R(u_k)||u_k||^n. The cited Hölder regularity theorems then apply directly, since each f_k is a bounded nonnegative continuous density on a strictly pseudoconvex domain. The remaining steps are internally consistent: Corollary 3.5 uses the comparison principle with the normalized eigenfunction w to get uk ≤ w < 0 and uniform Rayleigh-quotient bounds; Lemma 4.1 supplies continuity of E, Iµ, and R along the relevant subsequences; Lemma 4.2 identifies any cluster point as an eigenfunction via equality in Hölder; Lemma 4.3 uses interleaved subsequences and monotonicity to force a single limit. The two small issues I noticed — the possible non-membership of u0 in E1 when u0|∂Ω<0, and the L^p exponent typo — are cosmetic and do not change the conclusion. Thus I agree with the reader's ACCEPT verdict, and no adjustment is needed.","tokens_in":10747,"tokens_out":32958,"duration_ms":384603,"concrete_test":"With the corrected Hölder estimate, write out the L^{1+1/n} bound explicitly: for f_k = R(u_k)(-u_k)^n f, show ||f_k||_{L^{1+1/n}(dV)} ≤ ||f||∞^{1/n} R(u_0)||u_0||^n_{L^{n+1}(µ)} < +∞, then confirm that [GKZ08]/[Ch15] deliver a uniform C^α bound with this constant. This verifies the one step on which the Arzelà-Ascoli conclusion rests.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof line by line, I do not find a load-bearing gap. The uniform Hölder estimates (3.6) are the most delicate point: they require f_k = R(u_k)(-u_k)^n f to have uniformly bounded L^{1+1/n} norm. The displayed estimate in §3.5 has a harmless exponent typo — f^{1+1/n} ≤ ||f||∞^{1/n} f gives the factor ||f||∞^{1/n}, not ||f||∞^{1/(n+1)} — but the constant is still finite because R(u_k) and ||u_k||_{L^{n+1}(µ)} are controlled by the monotonicity quantity Q_k := R(u_k)||u_k||^n, which is non-increasing by Lemma 3.4. The resulting uniform C^α bounds are exactly the hypotheses of [Kol96], [GKZ08], and [Ch15], so the Arzelà-Ascoli step in §4.1 is supported. The comparison argument in Corollary 3.5, the Hölder-equality argument in Lemma 4.2, and the interleaving argument in Lemma 4.3 all check out. Two minor points are fixable: u0 with u0|∂Ω<0 need not lie in E1, so the invocation of Lemma 3.3 for R(u0) is slightly informal (though R(u0) is finite for Lipschitz psh data with finite total Monge-Ampère mass), and the L^p exponent typo mentioned above. Neither affects the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an iterative (inverse-iteration) scheme for the complex Monge-Ampère eigenvalue problem on a bounded strictly pseudoconvex domain. Given u0 ∈ PSH(Ω)∩C^{0,1}(Ωbar) with (dd^c u0)^n ≥ f dV and u0 ≤ 0 on ∂Ω, the iterates u_{k+1} are defined as the unique zero-boundary solutions to (dd^c u_{k+1})^n = R(u_k)(-u_k)^n f dV. The main theorem asserts that u_k converges uniformly on Ωbar to a nontrivial Hölder continuous plurisubharmonic eigenfunction u with eigenvalue λ1, and that R(u_k) → λ1^n. The proof combines a monotonicity lemma for the normalized Rayleigh quotient, uniform Hölder estimates for the iterates, Arzelà-Ascoli compactness, and uniqueness results from [BZ23].","tokens_in":11026,"tokens_out":32930,"duration_ms":367852,"significance":"The result is a natural constructive complement to the existence and uniqueness theory of [BZ23]. It transfers the Abedin-Kitagawa inverse iteration from the real to the complex Monge-Ampère setting and gives an effective scheme that approximates both the first eigenvalue and an eigenfunction without prior knowledge of λ1. The proof is clearly structured; the key steps that I checked are the monotonicity of R(u_k)||u_k||^n (Lemma 3.4), the uniform L^{1+1/n} bound on the right-hand sides (Section 3.5), the continuity lemma for the energy and Rayleigh quotient (Lemma 4.1), and the equality-in-Hölder argument identifying the limit (Lemma 4.2). The reliance on [BZ23] for existence, uniqueness, and the variational formula is transparent and does not make the algorithm circular.","major_comments":[],"minor_comments":[{"comment":"The sentence 'Observe that by Lemma 3.3, we have 0<R(u0)<∞' is not justified for the full class allowed in the Main Theorem, since u0 with u0|∂Ω<0 need not belong to E1. The finiteness nevertheless follows from boundedness of u0 and finiteness of its total Monge-Ampère mass; please add a short argument or restrict the statement to u0∈E1.","section":"2.2"},{"comment":"The proof asserts weak convergence of (-ϕ)(dd^cϕ_j)^n to (-ϕ)(dd^cϕ)^n without justification. This is true under the stated uniform convergence and uniform mass bound (e.g., via convergence in capacity), but a reference or a one-sentence proof should be supplied.","section":"4.1 (Lemma 4.1)"},{"comment":"The L^p estimate for f_k is correct as written: because the exponent is p=1+1/n, the factor is ||f||∞^{1/(n+1)}, not ||f||∞^{1/n}. Stating p explicitly would prevent confusion.","section":"3.5"},{"comment":"In Lemma 4.2, the reference to '(3.5) and (3.6)' for the uniform bound on (dd^c u_{k(j)})^n should also cite Corollary 3.5(ii), which bounds R(u_k); the current citation is slightly incomplete.","section":"4.2"},{"comment":"There are several typographical errors, including 'Aknowledgements', 'oper taor' in the reference [CLMcC24], and 'Koldziej' for 'Kołodziej'; these should be corrected.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The paper is essentially correct and the main theorem is well supported. I recommend minor revision. The stress-test note's claim of an exponent typo in Section 3.5 is not accurate: the displayed factor ||f||∞^{1/(n+1)} follows from p=1+1/n and is correct. The only substantive presentation point is the justification of finiteness of R(u0) for initial data not lying in E1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main new result is the uniform convergence of the inverse-iteration sequence to a first eigenfunction, with the Rayleigh quotients converging to λ1^n. That is a genuine extension of the Abedin–Kitagawa real result, and the complex adaptation is not a formality: the proof uses Cegrell classes, Bedford–Taylor comparison, and Kolodziej-type Hölder bounds. The structure is clear: the monotonicity of the normalized Rayleigh quotient is proved by a one-line Hölder inequality; the uniform Hölder estimates are invoked from the literature; Lemma 4.1 gives the continuity of the energy along uniformly convergent sequences with bounded mass, which is the delicate step; and the equality-in-Hölder argument forces the limit to satisfy the eigenfunction equation. The interleaving argument for full convergence of the sequence is correct.\n\nThe paper is honest about its scope: it approximates the eigenvalue/eigenfunction, relying on the BZ23 existence-uniqueness theorem for identification of the limit. It does not claim to reprove existence. The open question about whether the limit depends on the initial function is a useful pointer.\n\nSoft spots are minor. First, Lemma 3.3 is used to define R(u0), but for a general Lipschitz u0 with u0|∂Ω ≤ 0, the membership in E1 is not explicitly verified. The natural example u0 = Aρ is in E0, so the issue is only presentational, but a sentence or a restricted statement would remove it. Second, the displayed estimate in §3.5 may look as if it has an exponent error: it has ||f||∞^{1/(n+1)}, whereas the inequality f^{1+1/n} ≤ ||f||∞^{1/n} f suggests ||f||∞^{1/n}. The correct factor is indeed 1/(n+1) after raising to the power n/(n+1) in the L^{1+1/n} norm, so the line is fine. Third, the proof leans on deep external regularity results; that is acceptable in this field but means the paper is not self-contained.\n\nVerdict: the central argument holds up. I would send it to peer review. A referee familiar with pluripotential theory can check the cited Hölder theorems and the Cegrell class nuances; the rest is a clean adaptation. The paper will be a useful reference for iterative methods for complex Monge-Ampère equations.","headline":"A clean, correct extension of inverse iteration to the complex Monge-Ampère eigenvalue problem; convergence proof is sound, only minor presentational fixes needed.","tokens_in":11552,"tokens_out":7683,"would_cite":true,"duration_ms":80670,"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":"A fixed-point iteration on plurisubharmonic functions converges to an eigenfunction of the complex Monge-Ampère Dirichlet problem.","keywords":["complex Monge-Ampère operator","Dirichlet eigenvalue","inverse iteration","Rayleigh quotient","plurisubharmonic functions","Hölder regularity","strictly pseudoconvex domains"],"falsifier":"Run the iteration from two admissible initial functions on a domain whose first eigenfunction is explicitly known, such as the unit ball with $f\\equiv1$; if the two uniform limits are not positive multiples of the known eigenfunction, or if $R(u_k)$ does not converge to $\\lambda_1^n$, the main theorem would be false.","tokens_in":10522,"feed_emoji":"🔁","tokens_out":12572,"duration_ms":132383,"temperature":0.7,"pith_summary":"The paper proves that the Dirichlet eigenvalue problem for the complex Monge-Ampère operator can be solved by a simple inverse iteration. Starting from any plurisubharmonic function $u_0$ with $(dd^c u_0)^n \\ge f\\,dV$ and $u_0\\le0$ on the boundary, one repeatedly solves a standard Monge-Ampère Dirichlet problem with right-hand side $R(u_k)(-u_k)^n f\\,dV$, where $R$ is the Rayleigh quotient (energy divided by a weighted $L^{n+1}$ norm). The main theorem shows that the iterates converge uniformly on $\\bar{\\Omega}$ to a Hölder-continuous eigenfunction $u$, and that $R(u_k)\\to\\lambda_1^n=R(u)$, where $\\lambda_1$ is the first eigenvalue. This gives an effective, eigenvalue-free approximation scheme for both the first eigenvalue and its eigenfunction on bounded strictly pseudoconvex domains, extending the known real convex-domain iteration to the complex setting.","feed_headline":"Iteration scheme finds the first complex Monge-Ampère eigenvalue","feed_subtitle":"A fixed-point iteration on plurisubharmonic functions converges to the eigenfunction and computes the eigenvalue en route.","key_machinery":"The argument is carried by the inverse-iteration operator $T$ together with the Rayleigh quotient $R(\\phi)=E(\\phi)/I_\\mu(\\phi)$, where $E(\\phi)=\\frac1{n+1}\\int_\\Omega(-\\phi)(dd^c\\phi)^n$ is the Monge-Ampère energy and $I_\\mu(\\phi)=\\frac1{n+1}\\int_\\Omega(-\\phi)^{n+1}f\\,dV$. Each step solves the standard Dirichlet problem for the complex Monge-Ampère operator, and the engine of the proof is the monotonicity inequality $E(u_{k+1})/\\|u_{k+1}\\|_{L^{n+1}(\\mu)} \\le E(u_k)/\\|u_k\\|_{L^{n+1}(\\mu)}$, which keeps the Rayleigh quotients bounded, together with uniform Hölder estimates on the whole sequence. The comparison principle places every iterate below a fixed normalized eigenfunction, and the classical compactness principle for uniformly bounded equicontinuous families turns the uniform Hölder bound into a converging subsequence whose limit is then identified as the eigenfunction by uniqueness.","core_discovery":"The central claim is that the fixed-point operator $T$, defined by taking $\\psi=T(\\phi)$ to be the unique solution of $(dd^c\\psi)^n = R(\\phi)(-\\phi)^n f\\,dV$ with $\\psi=0$ on $\\partial\\Omega$, has the first eigenfunction as an attracting fixed point. The paper proves that for any admissible initial $u_0$ satisfying $(dd^c u_0)^n\\ge f\\,dV$ and $u_0\\le0$ on the boundary, the sequence $u_{k+1}=T(u_k)$ converges uniformly on $\\bar{\\Omega}$ to a limit $u\\in \\mathrm{PSH}(\\Omega)\\cap C^\\alpha(\\bar{\\Omega})$ for some $\\alpha\\in(0,1)$, with $u=0$ on $\\partial\\Omega$ and $u\\not\\equiv0$. Moreover $\\lim_k R(u_k)=\\lambda_1^n=R(u)$ and $(dd^c u)^n=(-\\lambda_1 u)^n f\\,dV$, so $u$ is an eigenfunction for the first eigenvalue $\\lambda_1$.","pith_inferences":["The same monotonicity-plus-compactness template is likely to transfer to complex Hessian operators, where an eigenvalue theory has recently been developed; one would only need a Hessian analogue of the uniform Hölder estimate and of the Rayleigh quotient.","The paper leaves open whether starting from the exact solution of $(dd^c u_0)^n=f\\,dV$ with zero boundary values forces the limit to be the particular normalized eigenfunction; if true, this would select a canonical representative of the eigenfunction and give the iteration a sharper convergence statement.","A quantitative version of the Hölder estimate (3.6) would turn the current convergence proof into an explicit convergence-rate result; the monotonicity inequality already provides a Lyapunov-type non-increasing quantity that could support such an estimate."],"forward_implications":["The first eigenvalue $\\lambda_1$ can be approximated without any prior knowledge of it, since the scheme produces a sequence $R(u_k)$ that converges to $\\lambda_1^n$.","Every limit produced by the scheme is an eigenfunction for the first eigenvalue; by the known uniqueness result, limits obtained from different admissible starting points are positive multiples of one another.","Each iterate solves a classical Dirichlet problem for the complex Monge-Ampère operator with continuous right-hand side, so the method can in principle be run with existing solvers and yields uniform convergence on the closed domain.","The Hölder exponent $\\alpha$ can be chosen in the explicit range $(0, 2/(n(n+1)+1))$, so the limiting eigenfunction has quantitative regularity.","The fixed-point reformulation connects the eigenvalue problem to inverse iteration methods familiar from linear algebra, giving a nonlinear analogue of Rayleigh-quotient iteration."],"supporting_citations":[{"why":"Established existence, uniqueness, and the Rayleigh-quotient formula for the first eigenvalue; the iteration targets this quantity.","marker":"[BZ23]"},{"why":"Supplied the inverse-iteration template in the real Monge-Ampère case that this paper adapts to strictly pseudoconvex domains.","marker":"[AK20]"},{"why":"Provides the Dirichlet existence theorem used to define the iterate $u_{k+1}$ from $u_k$.","marker":"[BT76]"},{"why":"Gives the uniform a priori estimate that keeps the sequence bounded.","marker":"[Kol96]"},{"why":"Provides Hölder regularity in a particular case used toward the uniform estimate (3.6).","marker":"[GKZ08]"},{"why":"Supplies the general Hölder regularity estimate that gives the compactness needed for convergence.","marker":"[Ch15]"},{"why":"Defines the energy class $E^1$ and the convergence properties on which the Rayleigh quotient and the energy functional rest.","marker":"[Ceg98]"}],"fun_headline_variants":["Iterative method converges to complex Monge-Ampère eigenfunction","Fixed-point iteration finds first Monge-Ampère eigenvalue","Convergent iteration for complex Monge-Ampère eigenvalue problem","New iteration computes complex Monge-Ampère eigenvalue","Iteration scheme converges to first Monge-Ampère eigenfunction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof rests on the uniform Hölder estimate (3.6) for the entire sequence of iterates: if the right-hand sides $R(u_k)(-u_k)^n f\\,dV$ failed to stay uniformly integrable at the required exponent, the sequence could lose compactness and the uniform limit and eigenfunction identification would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Iterative method converges to complex Monge-Ampère eigenfunction","Fixed-point iteration finds first Monge-Ampère eigenvalue","Convergent iteration for complex Monge-Ampère eigenvalue problem","New iteration computes complex Monge-Ampère eigenvalue","Iteration scheme converges to first Monge-Ampère eigenfunction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000221,"raw_usage":{"total_tokens":1393,"prompt_tokens":831,"completion_tokens":562,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":476}},"tokens_in":447,"tokens_out":562,"duration_ms":5469,"temperature":1.0,"reasoning_tokens":476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:26:50.371917+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the iteration from two admissible initial functions on a domain whose first eigenfunction is explicitly known, such as the unit ball with $f\\equiv1$; if the two uniform limits are not positive multiples of the known eigenfunction, or if $R(u_k)$ does not converge to $\\lambda_1^n$, the main theorem would be false.","supporting_citations":[],"review_version":1}