A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds

Liding Huang

REVIEW 4 major objections 5 minor 29 references

A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that a sub-slope inequality is necessary and sufficient for existence of smooth solutions to a broad class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, and derives applications to two…

desk verdict Plausible extension of Guo-Song sub-slope to almost Hermitian manifolds with new existence results for Hessian quotient and dHYM, but the presentation has several statement-level issues that need fixing. read the letter →

arxiv 2506.16305 v1 pith:FVF526LY submitted 2025-06-19 math.AP math.DG

Liding Huang This is my paper · ORCID

classification math.APmath.DG MSC 58J0532Q6035J60

keywords Fullynon-linearellipticequationssub-slopeC-subsolutionAlmostHermitianmanifoldComplexHessianquotientequationDeformedHermitian-Yang-MillsContinuitymethodAprioriestimates

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 establishes a general existence principle for fully nonlinear elliptic equations of the form $F(\omega_u)=h+\sigma$ on compact almost Hermitian manifolds, where $\omega_u$ is a deformed $(1,1)$-form depending on the unknown function and its gradient. The principle is that solvability is equivalent to a sub-slope condition: there exists an admissible function $u$ with $\sigma < \min(F_\infty(u)-h)$, where $\sigma$ is the infimum over admissible functions of the largest gap $F(u)-h$. The author generalizes the sub-slope method from Hermitian to almost Hermitian manifolds and drops the earlier positivity assumption on the right-hand side. As applications, the paper proves existence of smooth solutions for the complex Hessian quotient equation and for the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting. A reader would care because these equations are central in complex geometry, and the almost Hermitian case had resisted the previous existence arguments.

What carries the argument

The sub-slope is $\sigma=\inf_{u\in E}\max_M(F(u)-h)$, with $E$ the set of functions whose eigenvalue tuple lies in the cone $\Gamma$. The asymptotic function $f_\infty(\lambda)=\min_i \lim_{R\to\infty} f(\lambda_1,\dots,R,\dots,\lambda_n)$ characterizes C-subsolutions: $\bar u$ is a C-subsolution exactly when $f_\infty(\lambda(\omega_{\bar u}))>h$. The continuity method deforms the right-hand side by $h_t=(1-t)F(\bar u)+th$ and tracks a constant $c_t$; the sub-slope inequality ensures the subsolution property survives, the a priori estimate closes the set of good parameters, and the Fredholm alternative opens it. This machinery replaces the previously required positivity of the right-hand side and works for the deformed Hermitian-Yang-Mills phase equation as well.

What would settle it

A concrete check would be to take a compact almost Hermitian 4-manifold with data satisfying conditions (i)-(iii) and a genuine C-subsolution, then run the continuity path numerically: if second derivatives of $\varphi_t$ blow up while the sub-slope inequality holds, Theorem 3.1 would be false. Alternatively, a counterexample to the a priori estimate of [12] inside the dHYM or gradient-term class would collapse the closedness step directly.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: if there exists $u\in E$ with $\sigma<\min(F_\infty(u)-h)$, then the equation $F(\omega_u)=h+\sigma$ has a smooth solution $u\in E$. The proof runs a continuity path $F(\bar u+\varphi_t)=h_t+c_t$ between an already-solved equation for $\bar u$ and the target equation; the sub-slope inequality keeps $\bar u$ a C-subsolution at every step, the a priori estimates give uniform $C^{2,\alpha}$ control, and the Fredholm alternative gives openness. A later maximum-principle argument forces the final constant $c_t$ to equal the sub-slope $\sigma$. The paper also treats the deformed Hermitian-Yang-Mills equation, whose $f$ does not satisfy the growth condition (iii), by noting that the same subsolution-preservation argument goes through. The geometric consequences are Theorem 1.2 for the complex Hessian quotient equation and Theorem 1.3 for the deformed Hermitian-Yang-Mills equation.

Load-bearing premise

The whole existence result leans on the a priori $C^{2,\alpha}$ estimate for equations like (1.1) and (1.6) on compact almost Hermitian manifolds; if that estimate fails for some allowed data, the continuity-path closedness argument breaks.

Editorial extensions

If this is right

For data satisfying conditions (i)-(iii), the three statements in Theorem 1.1 become equivalent: a smooth solution exists, a C-subsolution exists, and the sub-slope inequality $\sigma<\min(F_\infty(u)-h)$ holds.
The complex Hessian quotient equation (1.4) on compact almost Hermitian manifolds admits a unique smooth solution whenever a C-subsolution exists.
The deformed Hermitian-Yang-Mills equation (1.6) with $h$ in the supercritical range admits a unique smooth solution on compact almost Hermitian manifolds whenever a C-subsolution exists.
The constant produced by the continuity method is forced by the maximum principle to be exactly the sub-slope $\sigma$, so the method solves the original equation with the threshold right-hand side, not merely some nearby constant.

Reading between the lines

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

The equivalence statement turns the existence problem into a check of one scalar inequality, so explicit almost Hermitian examples such as solvmanifolds could be tested for the sub-slope threshold against known obstructions.
Because the proof only needs the subsolution to survive along the path, the same scheme should transfer to other equations whose $f_\infty$ is bounded, including parabolic versions of these equations, once a $C^{2,\alpha}$ estimate is available.
Dropping the positivity of the right-hand side is likely to matter for geometric applications with indefinite background forms or sign-changing $h$, where earlier subsolution arguments did not apply.
The paper's treatment of gradient terms through $Z(\partial u)$ suggests the result may cover more general Lagrangian-type equations on almost complex manifolds if they can be cast in this form.

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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

Here's my take. The genuinely new contribution is the transfer of Guo-Song's sub-slope technique to compact almost Hermitian manifolds, including gradient terms, and the first existence statements for the complex Hessian quotient equation and dHYM in that setting. If the machinery works, that's a real gap filled.

The main sufficiency theorem (Theorem 3.1) follows the familiar continuity method: define sub-slope as an infimum, prove the subsolution is preserved along the path, then get closedness from the cited a priori estimates. The argument is concise and mostly standard.

The soft spots are real but mostly presentational. First, Theorem 1.1 states an equivalence but only proves (3)⇒(1). The reverse direction is not automatic—a solution of F(u)=h+σ need not satisfy the strict inequality σ < min(f∞(u)-h) that defines a C-subsolution. This overclaim should be corrected; the applications only need sufficiency.

Second, the definitions of σ in Theorems 1.2 and 1.3 (displayed in (1.5) and (1.8)) are not well-posed as written. The variable u' ranges over an unspecified set, and the expressions are missing the normalization by χ^n. These are fixable, but as written the statements are ill-defined.

Third, and more substantively, the closedness step in Section 3.1 invokes Theorem 2.4 on the solution φ_t while the C-subsolution is the fixed u. The theorem as quoted says 'u is a C-subsolution ... Suppose u is a smooth solution ...', which suggests the same function must play both roles. In the continuity argument they are different. If Huang-Zhang's estimate is of the standard separate-subsolution form, this is a misstatement; if not, the argument has a genuine gap. The paper should either quote the theorem correctly or prove that the solution is itself a C-subsolution along the path.

The dHYM application is only sketched; since dHYM does not satisfy condition (iii), the 'similar' proof deserves a few checks.

Overall, the core idea is sound and the paper deserves a serious referee. I would send it to review, but with a request for careful revision. In its current form I would not cite it, because the statements need fixing. After revision, it could be a useful paper for people working on fully nonlinear PDEs on almost complex manifolds.

Referee Report

4 major / 5 minor

Summary. The paper generalizes the sub-slope method of Guo-Song to compact almost Hermitian manifolds and claims an existence theorem for fully nonlinear elliptic equations of the form F(ω_u)=h+σ, including equations with gradient terms. The main result, Theorem 1.1, states an equivalence between solvability, existence of a C-subsolution, and a sub-slope inequality. The proof uses a continuity method, with closedness relying on a priori estimates quoted from Huang-Zhang and openness obtained by a Fredholm alternative argument. Applications are stated for the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting.

Significance. If the main existence theorem is correct, the paper gives a practically checkable criterion for solvability of a broad class of fully nonlinear equations on almost Hermitian manifolds, and it would provide the first existence results for the complex Hessian quotient equation and the dHYM equation in that category. The paper is concise and builds on substantial prior work: the C-subsolution framework of Székelyhidi, the sub-slope idea of Guo-Song, and the a priori estimates of Huang-Zhang. The overall strategy is natural and the openness argument is standard. However, several load-bearing points are not adequately supported in the manuscript as written: the quoted a priori estimate is applied in a form that does not match the continuity argument, the claimed equivalence in Theorem 1.1 is only proved in one direction, and the dHYM application is dismissed with a 'similar' argument despite failing one of the structural hypotheses. These issues are local and likely fixable, so the paper merits revision rather than rejection.

major comments (4)

[Theorem 1.1 and Section 3] The equivalence in Theorem 1.1 is not established. Section 3 proves (2) equivalent to (3) via Proposition 2.2 and then proves Theorem 3.1, which is the implication (3) to (1). The converse implication (1) to (3) is never proved. Lemma 3.7 only shows that a solution u of F(u)=h+σ satisfies σ=max_M(F(u)-h); this is an identity for a solution, not a proof that F∞(ω_u)>h+σ. If the intended argument is that every solution is automatically a C-subsolution, that fact should be stated and proved, for example by showing f∞>f on Γ under assumptions (i)-(iii) and for the dHYM phase. Otherwise Theorem 1.1 should be reformulated as a sufficiency theorem.
[Section 3.1, application of Theorem 2.4] The closedness of the continuity set applies Theorem 2.4 to φ_t, but the theorem as quoted on p.6 requires the same function to be both a C-subsolution and a solution. In the continuity path (3.5), the fixed function u is shown in Proposition 3.5 to be a C-subsolution, while the function whose C^{2,α} norm is needed is φ_t, or equivalently \bar u+φ_t. If Huang-Zhang [12, Cor. 1.4/Prop. 3.11] is the standard separate-subsolution estimate, then Theorem 2.4 must be restated with two distinct functions and the hypotheses for the deformed family (3.5) must be verified, including the gradient terms and the dHYM case. If the estimate really requires the solution itself to be a C-subsolution, then the bound \|φ_t\|\le C does not follow and the closedness step collapses.
[Theorems 1.2 and 1.3, equations (1.5) and (1.8)] The constants appearing in the two applications are not stated in the same normalization as Definition 2.2. In Theorem 1.2 the equation (1.4) contains σ in the exponent, while (1.5) defines log σ as the infimum of max(log(ω_{u'}^k∧χ^{n-k}/(ω_{u'}^l∧χ^{n-l}))-h); thus the σ in (1.4) is the exponential of the sub-slope of Definition 2.2, not the sub-slope itself. In Theorem 1.3, (1.8) defines tan σ, again a different normalization, and the minimum is taken over a restricted phase interval. If these formulas are intended as explicit computations of the sub-slope for the two equations, the equivalence should be stated and proved; otherwise the constant in the existence theorem is ambiguous.
[Theorem 1.3 and Section 3] The dHYM case is treated by the sentence 'the proof is similarly to Theorem 1.1' (p.4), but Theorem 3.1 and its proof rely on condition (iii), on the sub-slope definition through inf max(F-h), and on the strict inequality in (3.4). The dHYM phase function does not satisfy condition (iii), and the sub-slope in (1.8) is defined through tan and a restricted phase interval rather than through Definition 2.2. The manuscript should either carry out the dHYM adaptation lemma by lemma, or explicitly identify which of the claims (3.3)-(3.8) remain valid and why. As written, Theorem 1.3 is not proved.

minor comments (5)

[Title and page 2] The title reads 'FULL Y NON-LINEAR' and should read 'FULLY NON-LINEAR'; there are also typos such as 'seting' on p.3 and 'SUpposethat' on p.8.
[Introduction, references] In the first paragraph the citation list '[5, 6, 12, 11, 12]' contains duplicate [12] and is out of order; please correct.
[Lemma 3.7] Lemma 3.7 refers to 'the sup-slope' although Definition 2.2 defines the sub-slope; either the terminology should be consistent or the intended distinction should be explained.
[Section 3, proof of Theorem 1.2] The introduction states that Theorem 1.2 will be proved in Section 3, but Section 3 ends after the lemmas for Theorem 3.1; a short verification that the Hessian quotient equation satisfies conditions (i)-(iii) and that (1.5) is the corresponding sub-slope should be added.
[Theorem 2.4 and notation] The letter u is overloaded: in Theorem 2.4 it denotes both the subsolution and the solution, and in Section 3.1 it denotes the fixed admissible function while the solution is \bar u+φ_t. Using distinct symbols such as \underline u and v would prevent the ambiguity that is currently at the center of the closedness argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sub-slope is defined as an infimum and the existence proof proceeds by a continuity method whose key a priori estimates are prior published inputs, not consequences of the target theorem.

full rationale

The paper's central claim, Theorem 3.1, asserts that the condition σ < min(F∞(u) − h) implies a smooth solution of F(u) = h + σ. The constant σ is defined as an infimum, σ = inf_{u∈E} max_M(F(u) − h), and is not fitted to any solution; Lemma 3.7 merely identifies the constant forced by the maximum principle with this infimum, which is a sharpness statement rather than a circular reduction. The continuity method is standard: a subsolution is preserved along the path, a priori estimates give closedness, and the implicit function theorem gives openness. The only input that could raise circularity concerns is the use of the author's prior work, Huang–Zhang [12], for Theorem 2.4 (the C^{2,α} a priori estimates). That citation is load-bearing for the closedness step, but it is an independent, parameter-free estimate whose stated assumptions do not include the target existence result; it is not a renaming of the theorem being proved, nor does it define σ. There is a possible correctness gap in that the paper's quoted Theorem 2.4 appears to require the same function to be both the C-subsolution and the solution, whereas the closedness argument only shows the fixed function u is a C-subsolution of the deformed family. If the underlying estimate in [12] is of the separate-subsolution form, this is a misstatement; if not, the closedness argument would be invalid. Either way, this is a correctness or citation-accuracy issue, not circularity, because it does not make the conclusion equivalent to its input by construction. Accordingly, no circular step can be exhibited with the required specificity, and the appropriate finding is no significant circularity.

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

The proof rests on previously established a priori estimates and subsolution characterizations for fully nonlinear equations on almost Hermitian manifolds; no empirical free parameters or new mathematical entities are introduced. The sub-slope sigma is defined as an infimum, not fitted to data.

assumptions (5)

domain assumption C^{2,alpha} a priori estimates for solutions with a C-subsolution on compact almost Hermitian manifolds (Theorem 2.4)
Invoked in Section 3.1 to obtain uniform bounds on phi_t and establish closedness of the continuity path; cited to Huang-Zhang [12, Corollary 1.4 or Proposition 3.11].
domain assumption C-subsolution characterization via f_infinity (Proposition 2.2)
Used to convert the sub-slope condition sigma < min(F_infinity(u)-h) into existence of a C-subsolution for the continuity family (3.5); cited to Szekelyhidi [25] and Collins-Jacob-Yau [3].
domain assumption Properties of f_infinity (Proposition 2.3)
Used in the chain of inequalities in Proposition 3.4 and in the characterization of C-subsolutions; cited to Trudinger [26], Szekelyhidi [25], and Guo-Song [8].
standard math Standard elliptic theory: maximum principle, Fredholm alternative, inverse function theorem, Schauder regularity
Used throughout Section 3 for the maximum principle comparisons and the openness argument.
domain assumption For dHYM, the C-subsolution characterization of Proposition 2.1 and the validity of Proposition 2.2 for dHYM (Remark 2.1)
Used to treat the dHYM equation in the same continuity framework despite failure of condition (iii); cited to Collins-Jacob-Yau [3].

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Pith. "Pith review of A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds." pith.science (2026). https://pith.science/paper/FVF526LY

@misc{pith2026250616305,
  author       = {Pith},
  title        = {Pith review of: A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVF526LY}},
  note         = {Machine review of arXiv:2506.16305}
}

read the original abstract

In this paper, we generalize the definition of sub-slope, introduced by Guo-Song, to almost Hermitian manifolds and prove the existence of solutions for a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting.

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Pith tools

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

[1] [1]

The Dirichlet problem for nonlinear second- order elliptic equations

Caﬀarelli, L., Nirenberg, L., Spruck, J. The Dirichlet problem for nonlinear second- order elliptic equations. III. Functions of the eigenvalue s of the Hessian , Acta Math. 155 (1985), no. 3-4, 261–301

work page 1985

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Calabi, E. On K¨ ahler manifolds with vanishing canonical class , Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 78–89 . Princeton University Press, Princeton, N. J., 1957

work page 1957

[3] [3]

(1,1) forms with speciﬁed Lagrangian phase: a priori estimates and algebraic obstructions, Camb

Collins, T., Jacob, A., Yau, S-T. (1,1) forms with speciﬁed Lagrangian phase: a priori estimates and algebraic obstructions, Camb. J. Math. 8 (2020), no. 2, 407–452

work page 2020

[4] [4]

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Chu, J., McCleerey, N. Fully non-linear degenerate elliptic equations in complex ge- ometry, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109176, 45 pp

work page 2021

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Zhang, J

Chu, J; Huang, L. Zhang, J. Fully non-linear elliptic equations on compact almost Hermitian manifolds, Calc. Var. Partial Diﬀerential Equations 62 (2023), no. 3

work page 2023

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The Monge–Amp` ere equation for non-integrable almost complex structures , J

Chu, J., Tosatti, V., Weinkove, B. The Monge–Amp` ere equation for non-integrable almost complex structures , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1949–1984

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Fang, H., Lai, M., Ma, X.-N. On a class of fully nonlinear ﬂows in K¨ ahler geometry , J. Reine Angew. Math. 653 (2011), 189–220

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arXiv