Power Convexity of Solutions to the Complex Monge-Amp\`{e}re Equation $\det(u_{i\overline{j}})=1$ in Complex Dimension Two

Hongyu Chen; Jingchen Hu; Li Sheng

Power Convexity of Solutions to the Complex Monge-Amp\`{e}re Equation $\det(u_{i\overline{j}})=1$ in Complex Dimension Two

T0 review · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Square root of negative solution is convex in complex dimension 2

desk verdict Power convexity of complex Monge-Ampère solutions in C²: correct result, intricate proof, one notational gap in the key inequality read the letter →

arxiv 2607.06895 v1 pith:BM55M5OH submitted 2026-07-08 math.AP

Hongyu Chen , Jingchen Hu , Li Sheng This is my paper · ORCID

classification math.AP

keywords complexconvexitydimensionequationmonge-ampoverlinepowersolutions

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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 studies the Dirichlet problem for the complex Monge-Ampère equation det(u_{i⎵j})=1 on a strictly convex bounded domain in C² with zero boundary data. The solution u is itself generally not convex. The authors prove that the transformed function −√(−u) is strictly convex throughout the domain. The key mechanism is an auxiliary function σ, built entirely from complex derivatives, that encodes the real convexity of −√(−u). Reducing convexity to positivity of σ, the authors derive a differential inequality for σ and then verify that the resulting quadratic form—a skew-Hermitian trace-minus-norm expression—is non-positive specifically in complex dimension 2, which is the unique step where the dimension restriction enters.

What carries the argument

Auxiliary eigenvalue function σ = λ_min(MA⁻¹) encoding real convexity via complex derivatives; differential inequality for σ derived from second-derivative relations of the Monge-Ampère equation; non-positivity of the quadratic form |tr(W)|² − Σ|W_{p⎵s}|² under the constraint Σ W_{p⎵q} ū^{qp} = 0 in complex dimension 2; perturbation by pluriharmonic quadratic polynomials to handle eigenvalue multiplicity; continuation from a ball to a general strictly convex domain.

What would settle it

Construct a smooth strictly convex domain in C² and a solution to det(u_{i⎵j})=1 with zero boundary data for which −√(−u) fails to be convex at some interior point—i.e., exhibit a point where the real Hessian of −√(−u) has a negative eigenvalue. Alternatively, find an error in the reduction from real Hessian positivity to the auxiliary function σ, or in the non-positivity proof of the quadratic form in dimension 2.

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Extended reading notes

Core claim

For the complex Monge-Ampère equation det(u_{i⎵j})=1 in a strictly convex domain in C² with zero Dirichlet boundary data, the function −√(−u) is strictly convex. This is established by introducing an auxiliary quantity σ (the minimum eigenvalue of a matrix K = M·A⁻¹ constructed from complex second derivatives of u) that measures convexity of −√(−u), deriving a differential inequality u^{p⎵q} σ_{p⎵q} ≤ C(σ + |∇σ|), and proving that the key quadratic form |tr(W)|² − Σ|W_{p⎵s}|² is non-positive under the constraint Σ W_{p⎵q} ū^{qp} = 0—a fact that holds in complex dimension 2 but fails in higher dimensions. A perturbation and continuation argument extends the estimate from the case of distinct,

Load-bearing premise

The entire proof hinges on the algebraic fact that the quadratic form |tr(W)|² − Σ|W_{p⎵s}|² is non-positive for skew-Hermitian matrices W satisfying a linear constraint, which the authors verify only in complex dimension 2 and explicitly show fails in dimensions 3 and higher. If this inequality fails, the differential inequality for σ breaks down and the maximum-principle argument cannot proceed.

Editorial extensions

If this is right

The power-convexity result with exponent α = 1/2 is now established for complex Monge-Ampère in C², raising the natural question of which exponents α ∈ (0,1) yield convexity of −(−u)^α.
The authors note their approach applies in all dimensions except for the single algebraic step (non-positivity of the quadratic form), so identifying a replacement argument in higher dimensions is the direct next problem.
The method of encoding real Hessian convexity through complex-derivative auxiliary functions may extend to other fully nonlinear elliptic equations where real-derivative computations are intractable.
The continuation argument from a ball to a general strictly convex domain via the family Ω_t = tΩ + (1−t)B provides a template for transferring convexity results on model domains to general domains.

Reading between the lines

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

The dimension-2 restriction is sharp for the specific quadratic form used, but the authors' claim that the approach works in all dimensions suggests that a modified auxiliary function or a different algebraic identity might recover the result in higher dimensions.
If power convexity of −√(−u) holds, one might expect that the level sets of u are also convex (or at least have convexity properties), since convexity of a monotone transform of u implies geometric constraints on its sublevel sets.
The perturbation technique—testing convexity of −√(−u−q) for all small pluriharmonic quadratic polynomials q—is reminiscent of viscosity-solution methods and may connect to stability questions for the Monge-Ampère equation under domain perturbations.

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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 result is that −√(−u) is strictly convex for solutions of det(u_{i⎵j})=1 on strictly convex domains in C². This is a real theorem — independently confirmed by Zhang-Zhou [ZZ26] via a different method, which the authors acknowledge. The technique is the genuinely interesting part: they introduce an auxiliary function σ built from complex derivatives to quantify real Hessian positivity, reducing the problem to a differential inequality for the minimal eigenvalue of MA⁻¹. The deformation argument connecting a ball to a general domain (Section 3.3) is clean and standard. The dimension-2 restriction is honestly identified and justified — Remark 2 explicitly shows the key algebraic inequality fails for n>2. That's good scholarship. The algebra lemmas in the appendix are parameter-free and check out. The soft spot is in Section 3.2. Equation (3.97) decomposes u^{p⎵q}∂_{p⎵q}σ into three terms: |tr(W)|² − Σ|W_{p⎵s}|² (Terms 1–2) and −2Σ H_{ps}H_{qs}/μ_s (Term 3). Section 3.2 proves Terms 1–2 are non-positive in dimension 2. Term 3 is non-positive if H_{ps}H_{qs} carries implicit conjugation (giving h*U⁻¹h ≥ 0, automatic). But the notation is ambiguous — Q₃ in (3.53) involves ∂_p M_{1⎵s} · ∂_{⎵q} M_{s⎵1}, which by Hermiticity of M does carry conjugation. The simplification to (3.93) writes H_{ps}H_{qs} without making the conjugation explicit, and H_{ps} = ∂_p M_{s1} (eq. 3.92) is a different matrix entry than what appears in Q₃. Remark 3 only shows Term 3 vanishes in the special case b_s=0, not that it's always ≤ 0. I think this is a notational gap, not a substantive error — the conjugation structure is likely preserved through the computation, and the independent proof by Zhang-Zhou supports correctness. But the paper should state this explicitly. The computation in Section 3.1 is also very long (pages of index manipulation), which makes verification tedious but not impossible. This is a paper for specialists in geometric PDE and complex Monge-Ampère equations. The methodological framework — using complex derivatives to control real convexity — is worth seeing even if the result itself is not unique. It deserves a serious referee who can check the index algebra line by line and confirm the conjugation structure in Term 3.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the proof is a self-contained PDE argument with parameter-free algebraic lemmas from self-citations

full rationale

The paper proves power convexity of solutions to the complex Monge-Ampère equation via a differential inequality for the minimal eigenvalue σ of K = MA^{-1}. The derivation chain is: (1) derive differential relations for A, B, M (Section 2), (2) compute u^{p̄q}∂_{p̄q}σ as a quadratic form in W and H (Section 3.1, culminating in eq. 3.97), (3) prove non-positivity of this quadratic form in complex dimension 2 (Section 3.2), (4) extend via perturbation (Section 3.3). The self-cited works [H25], [H24B], [H24A], [HS26] provide Lemma A.1 (an algebraic identity det(I - BA^{-1}BA^{-1}) = det D / (4^n (det A)^2), verified by direct computation in coordinates) and Lemma A.4 (a convexity estimate). These are parameter-free algebraic facts, not fitted results or ansätze. The deformation argument connecting a ball to Ω (eq. 3.134) is standard. The dimension-2 restriction is explicitly traced to the algebraic inequality |tr(W)|² - Σ|W_{p̄s}|² ≥ 0 under constraint (3.99), verified in Section 3.2 and shown to fail for n > 2 (Remark 2). The skeptic's concern about Term 3 in (3.97) is a correctness/notation issue (whether the quadratic form h^*U^{-1}h has proper conjugation), not a circularity issue — the term is derived, not defined to equal its input. The independent verification by [ZZ26] via a different method provides external support. No step reduces to its inputs by construction, no prediction is a renamed fit, and no uniqueness theorem is invoked to forbid alternatives. The self-citations are load-bearing but independent (algebraic identities), raising the score only to 2 for minor self-citation dependence on prior algebraic lemmas that could be verified independently of the present paper's main result.

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

No free parameters, no ad hoc axioms, no invented entities. The proof is a self-contained PDE argument relying on standard results from complex Monge-Ampère theory and linear algebra.

assumptions (4)

standard math Existence and regularity of classical solutions u ∈ C⁴(Ω) ∩ C²(Ω̄) to the Dirichlet problem det(u_{i⎵j})=1 on strictly convex smooth domains
Invoked in Problem 1 and Theorem 1.1; standard from complex Monge-Ampère theory.
standard math C² estimates for Problem 1 giving positive lower bound on minimum eigenvalue of (u_{i⎵j}) in Ω
Stated in Section 2.2: 'According to the C² estimate for Problem 1, the minimum eigenvalue of (u_{i⎵j}) has a positive lower bound estimate in Ω.'
standard math Strict convexity of −√(−u) near the boundary (in Ω_δ ∖ Ω_{2δ}) for smooth strictly convex domains
Section 2.2: 'it's well known that v is strictly convex in Ω_δ ∖ Ω_{2δ} when δ is small enough.' Used as boundary condition for the interior argument.
standard math Lemma A.6 of [H25] / Lemma A.1: equivalence of real convexity to positivity of A and M = A − BA⁻¹B
Used throughout to translate real Hessian positivity into complex matrix conditions. Proved here as Lemma A.1 for quadratic polynomials.

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Pith. "Pith review of Power Convexity of Solutions to the Complex Monge-Amp\`{e}re Equation $\det(u_{i\overline{j}})=1$ in Complex Dimension Two." pith.science (2026). https://pith.science/paper/BM55M5OH

@misc{pith2026260706895,
  author       = {Pith},
  title        = {Pith review of: Power Convexity of Solutions to the Complex Monge-Amp\`ere Equation $\det(u_i\overlinej)=1$ in Complex Dimension Two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BM55M5OH}},
  note         = {Machine review of arXiv:2607.06895}
}

abstract

In this paper, we establish the power convexity of solutions to the complex Monge-Amp\`{e}re equation $\det(u_{i\overline{j}})=1$ in a convex set in the complex 2 dimensional Euclidean space. Even the convexity result is only for 2 dimension, our approach is applicable to all dimensions.

Figures

Figures reproduced from arXiv: 2607.06895 by the authors.

**Figure 2.** Graph of σ ϵ is the Lower Envelop of the Graph of λ1(Kq ) with q ∈ Qϵ At any point z0 ∈ Ωε , since Qϵ is closed and of finite dimension, we can find r ∈ Qϵ so that λ1(K r )(z0) = σ ϵ (z0), (3.130) and for z ̸= z0 λ1(K r )(z) ≥ σ ϵ (z). (3.131) This means the graph of λ1(Kr ) contacts with the graph of σ ϵ from above. In addition, Lemma A.3 implies λ1(K r )(z0) < λ2(K r )(z0). (3.132) So λ1(Kr ) is a smooth function … view at source ↗

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.

[1] [1]

B. Bian, P. Guan, A microscopic convexity principle for nonlinear partial differential equations , Invent. math. 177, 307–335 (2009)

work page 2009

[2] [2]

B. Bian, P. Guan, X. Ma, L. Xu, A constant rank theorem for quasiconcave solutions of fully nonlinear partial differential equations , Indiana University Mathematics Journal, Vol. 60, No. 1 (2011), pp. 101-119

work page 2011

[3] [3]

Caffarelli, P

L. Caffarelli, P. Guan, X. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations , Communications on Pure and Applied Mathematics 60(12):1769 - 1791

work page

[4] [4]

Caffarelli, A

L. Caffarelli, A. Friedman, Convexity of Solutions of Semilinear Elliptic Equations, Duke Math. J. 52(2): 431-456 (June 1985)

work page 1985

[5] [5]

C, Chen, H, Jia, J, Xiong, Strict Power Convexity of Solutions to Fully Nonlinear Elliptic Partial Differential Equations in Two Dimensional Convex Domains . J. Differential Equations (2025) 419:505--517

work page 2025

[6] [6]

C. Chen, J. Li, X. Ma, Brunn--Minkowski Inequality for the First Complex -Hessian Eigenvalue, arXiv:2606.25678

work page Pith review arXiv

[7] [7]

Xu, Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations , J

P.Guan, L. Xu, Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations , J. Reine Angew. Math.vol. 2013, no. 680, 2013, pp. 41-67

work page 2013

[8] [8]

R. Horn, C. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, Second Edition 2013

work page 2013