An Iterative Approach to the Complex Monge-Amp\`ere Eigenvalue Problem

Ahmed Zeriahi

An Iterative Approach to the Complex Monge-Amp\`ere Eigenvalue Problem

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Pith's one-line read A fixed-point iteration on plurisubharmonic functions converges to an eigenfunction of the complex Monge-Ampère Dirichlet problem.

desk verdict A clean, correct extension of inverse iteration to the complex Monge-Ampère eigenvalue problem; convergence proof is sound, only minor presentational fixes needed. read the letter →

arxiv 2507.13273 v1 pith:BHHEYK56 submitted 2025-07-17 math.CV math.AP

Ahmed Zeriahi This is my paper · ORCID

classification math.CVmath.AP MSC 31C4532U1532U4032W2035J6635J96

keywords complexMonge-AmpèreoperatorDirichleteigenvalueinverseiterationRayleighquotientplurisubharmonicfunctionsHölderregularitystrictlypseudoconvexdomains

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The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\lambda_1$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

Soft 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.

Verdict: 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.

Referee Report

0 major / 5 minor

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].

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.

minor comments (5)

[2.2] 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.
[4.1 (Lemma 4.1)] 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.
[3.5] 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.
[4.2] 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.
[Throughout] There are several typographical errors, including 'Aknowledgements', 'oper taor' in the reference [CLMcC24], and 'Koldziej' for 'Kołodziej'; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the iterative scheme is self-contained relative to the external existence/uniqueness theory, and the limit identification uses [BZ23] only as an independent anchor.

full rationale

The paper's goal is to approximate, not to re-prove, the known eigenpair. The fixed-point reformulation in equations (2.1)-(2.2) is a genuine equivalence: if (dd^c u)^n = (-λ u)^n µ, then λ^n = R(u), and conversely any fixed point of T solves the eigenvalue equation with λ = R(u)^{1/n}. This is a reformulation, not a circular reduction. The iteration (2.4) is defined entirely in terms of the current iterate u_k and the Rayleigh quotient R(u_k); it never uses the value of λ_1 or the normalized eigenfunction w. The comparison with w in Corollary 3.5 uses the existence result of [BZ23] only to establish the uniform bounds needed for compactness; the bounds do not enter the construction of the sequence. The uniform Hölder estimates (3.6) are proved from an L^{1+1/n} bound on the right-hand side f_k and are cited to [Kol96], [GKZ08], and [Ch15], which are external regularity theorems independent of the target conclusion. The uniqueness statement from [BZ23] is invoked only to identify the limit point produced by the Arzelà-Ascoli argument, not to force the iteration's outcome. No data are fitted, no quantity is renamed as a parameter, and no load-bearing premise is justified solely by a self-citation. The paper openly builds on prior work by the same author, but that reliance is appropriate and does not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's central theorem rests on the prior existence and uniqueness of the complex Monge-Ampere eigenvalue problem and on standard but deep pluripotential estimates. No new free parameters or invented entities are introduced.

assumptions (4)

domain assumption Existence and uniqueness of the Dirichlet eigenvalue problem (Theorem A of BZ23).
The paper uses the normalized eigenfunction w with (dd^c w)^n = (-lambda1 w)^n f dV to construct comparison functions in Corollary 3.5 and to identify the limit in Lemma 4.2.
domain assumption Uniform Holder a priori estimates for solutions to (dd^c u)^n = g dV with g in L^{1+1/n} (from Kol96, GKZ08, Ch15).
Needed in Section 3.5 to get the uniform bound (3.6) used for compactness in Section 4.1.
standard math Bedford-Taylor comparison principle and existence of continuous solutions to (dd^c u)^n = h dV when h is continuous and nonnegative (BT76).
Used to define the operator T in (2.2) and to compare uk and w in Corollary 3.5.
standard math Cegrell's theory of the classes E0 and E1, including approximation and integration by parts (Ceg98).
Underpins the definitions of E and I_mu and the proof of Lemma 3.3.

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Pith. "Pith review of An Iterative Approach to the Complex Monge-Amp\`ere Eigenvalue Problem." pith.science (2026). https://pith.science/paper/BHHEYK56

@misc{pith2026250713273,
  author       = {Pith},
  title        = {Pith review of: An Iterative Approach to the Complex Monge-Amp\`ere Eigenvalue Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BHHEYK56}},
  note         = {Machine review of arXiv:2507.13273}
}

abstract

We present an iterative approach to approximate the solution to the Dirichlet complex Monge-Amp\`ere eigenvalue problem on a bounded strictly pseudoconvex domain in $\C^n$. This approach is inspired by a similar approach initiated by F. Abedin, J. Kitagawa who considered the real Monge-Amp\`ere operator on a strictly convex domain in $\R^N$.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Abedin, J

F. Abedin, J. Kitagawa : Inverse iteration for the Monge-Amp\`ere eigenvalue problem . Proceedings AMS, 148 (2020), no 11, 4875-4886

work page 2020
[2]

P. Åhag, U. Cegrell, R. Czyz : On Dirichlet's principle and problem. Math. Scand. 110 (2), 235-250 (2012)

work page 2012
[3]

Badiane, A

P. Badiane, A. Zeriahi : The eigenvalue problem for the complex Monge-Amp\`ere opertaor . J. Geom. An. (2023), 33:367, 44pp

work page 2023
[4]

Bedford, B

E. Bedford, B. A. Taylor : The Dirichlet problem for a complex Monge-Amp\`ere equation. Invent. Math., 37 (1), 1-44 (1976)

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[5]

Bedford, B

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Berestycki, L

H. Berestycki, L. Nirenberg, S.R.S. Varadhan : The first eigenvalue and maximum principle for second order elliptic differential operators in general domains. Comm. Pure Appl. Math. 47 (1), 47-92 (1994)

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[7]

Berman, S

R. Berman, S. Boucksom, V. Guedj, A. Zeriahi : A variational approach to complex Monge-Amp\`ere equations. Publ. Math. Inst. Hautes Études Sci. 117, 179-245 (2013)

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[8]

Caffarelli, J

L. Caffarelli, J. J. Kohn, L. Nirenberg, J. Spruck : The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Amp\`ere, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (2), 209-252 (1985)

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J. Chu, Y. Liu, N. McCleerey : The eigenvalue problem for the complex Hessian opertaor on m psuedoconvex manifolds . Preprint arXiv:2402.03098v1

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V. Guedj, S. Koldziej, A. Zeriahi : Hölder continuous solutions to Monge-Amp\`ere equations. Bull. Lond. Math. Soc. 40 (6), 1070–1080 (2008)

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Abedin, J

F. Abedin, J. Kitagawa : Inverse iteration for the Monge-Amp\`ere eigenvalue problem . Proceedings AMS, 148 (2020), no 11, 4875-4886

work page 2020

[2] [2]

P. Åhag, U. Cegrell, R. Czyz : On Dirichlet's principle and problem. Math. Scand. 110 (2), 235-250 (2012)

work page 2012

[3] [3]

Badiane, A

P. Badiane, A. Zeriahi : The eigenvalue problem for the complex Monge-Amp\`ere opertaor . J. Geom. An. (2023), 33:367, 44pp

work page 2023

[4] [4]

Bedford, B

E. Bedford, B. A. Taylor : The Dirichlet problem for a complex Monge-Amp\`ere equation. Invent. Math., 37 (1), 1-44 (1976)

work page 1976

[5] [5]

Bedford, B

E. Bedford, B. A. Taylor : A new capacity for plurisubharmonic functions. Acta Math. 149 , no. 1 (2), 1-40 (1982)

work page 1982

[6] [6]

Berestycki, L

H. Berestycki, L. Nirenberg, S.R.S. Varadhan : The first eigenvalue and maximum principle for second order elliptic differential operators in general domains. Comm. Pure Appl. Math. 47 (1), 47-92 (1994)

work page 1994

[7] [7]

Berman, S

R. Berman, S. Boucksom, V. Guedj, A. Zeriahi : A variational approach to complex Monge-Amp\`ere equations. Publ. Math. Inst. Hautes Études Sci. 117, 179-245 (2013)

work page 2013

[8] [8]

Caffarelli, J

L. Caffarelli, J. J. Kohn, L. Nirenberg, J. Spruck : The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Amp\`ere, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (2), 209-252 (1985)

work page 1985