Remarks on Hessian quotient equations on Riemannian manifolds

Marcin Sroka

REVIEW 3 major objections 4 minor 27 references

Remarks on Hessian quotient equations on Riemannian manifolds

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A new maximum-principle test function yields unobstructed second-order estimates for the real Hessian quotient equation in dimension two.

desk verdict Genuinely new test-function idea and an honest paper, but the advertised proof has a load-bearing gap in Proposition 2.4 that needs fixing before the new argument can stand. read the letter →

arxiv 2501.03386 v1 pith:A5RQC53G submitted 2025-01-06 math.DG math.AP

Marcin Sroka This is my paper · ORCID

classification math.DGmath.AP MSC 58J0535R01

keywords Hessianquotientequationsfullynonlinearellipticsecond-orderaprioriestimatesmaximumprincipleRiemannianmanifoldspositiveoperatorsJ-equationMonge–Ampèreequation

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 proves that the real Hessian quotient equation $F(u)=\frac{\sigma_2}{\sigma_1}(\lambda(g^{-1}(\chi+\nabla^2 u)))=f$ is unobstructed in dimension two: every admissible solution on a closed connected Riemannian surface satisfies $|\nabla^2 u|_g\le C$, with $C$ depending only on $f$, the metric $g$, and the background tensor $\chi$. Such second-order bounds are the main missing step for solvability of Hessian quotient equations, a problem raised in the literature. The proof avoids the classical reduction to the two-dimensional Monge–Ampère interior estimate and instead runs a maximum principle on a new test quantity built from $\log\lambda_1$ plus a directional derivative along the eigenvector of the largest eigenvalue, exploiting an exact concavity identity for the quotient operator. From the bound, the paper derives existence and uniqueness of smooth admissible solutions for any right-hand side up to rescaling. This stands in contrast to the complex J-equation, where the same curvature operator has genuine obstructions even on complex surfaces.

What carries the argument

The load-bearing object is the new test quantity $\tilde Q(x)=\log\lambda_1(x)+\sup_v\phi(\frac{1}{2}u_v(x)^2)$, where $\lambda_1$ is the largest eigenvalue of $g^{-1}(\chi+\nabla^2 u)$, the supremum is over unit eigenvectors belonging to $\lambda_1$, and $\phi'$ is taken to be a large constant $A$. At the maximum, this is compared with $Q$ built from a local unit eigenvector field $V$; the extremal equation (3.21) and the explicit derivative formulas (3.22)–(3.29) recast all third-order terms into quadratic forms bounded by the $\mathrm{C}^1$ estimate. The second ingredient is Proposition 2.6, the exact concavity identity $-F^{ii,jj}\xi_i\xi_j=2F^{ii}\xi_i^2/\lambda_i-2(F^{ii}\xi_i)^2/F$ for $F=\sigma_n/\sigma_{n-1}$; it is what turns the linearized operator on $\log\lambda_1$ into the strictly positive term $F^{11}\tilde g_{11,1}^2/\lambda_1^2$. The restriction $n=2$ enters only when solving the $2\times2$ system that controls the derivatives of $\tilde g$ and $V$, which is where the proof would need new ideas in higher dimension.

What would settle it

Compare the asymptotics of both sides of (3.35) as $\lambda_1\to\infty$, substituting $F^{11}=F^2/\lambda_1^2$ and $V^2_1u_2+\tilde g_{11}-\chi_{11}\approx\lambda_1$; if the right-hand side is $O(\phi'^2|u_1|^2)$ rather than growing with $\lambda_1$, the claimed contradiction does not occur. This coefficient check is a finite computation using (2.9), (2.16), and the $\mathrm{C}^1$ bound of Proposition 5.2.

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Extended reading notes

Core claim

The central claim is Theorem 2.2: in real dimension two, any admissible solution of (2.3) has a uniform bound on the full Hessian, $|\nabla^2 u|_g\le C(f,g,\chi)$, with no subsolution, superslope, or curvature assumption. The discovery is that this second-order estimate is unconditional for the positive Hessian quotient operator $F=\sigma_2/\sigma_1$, and that the obstruction is absent where the complex analogue has one. The proof identifies a structural reason: writing $F=1/\sigma_1(\lambda^{-1})$ yields the sharp concavity identity (2.16), and the new test function (3.1) lets the maximum principle consume all third-order terms, leaving only the controllable positive term $F^{11}\tilde g_{11,1}^2/\lambda_1^2$. The result implies, by the continuity method, that a single admissible function is enough to solve (2.3) for any positive $f$ up to rescaling.

Load-bearing premise

The proof's final contradiction requires that the big positive term in its last estimate get larger as the largest Hessian eigenvalue gets larger; if that term stays the same size instead, nothing in the estimate stops the eigenvalue from going to infinity.

Editorial extensions

If this is right

In real dimension two, every admissible solution of (2.3) carries the a priori bound $|\nabla^2 u|_g\le C$, so the second-order estimate is unobstructed and the equation is solvable for any positive right-hand side $f$ up to rescaling.
No subsolution, superslope, or curvature assumption is needed for the bound; earlier conditional frameworks are bypassed in this case.
The normalized equation $F(u)=e^{\int_M u\,\mathrm{vol}_g+\Psi}$ has a unique smooth admissible solution whenever at least one admissible reference function exists.
The real situation is genuinely different from the complex J-equation, which fails to be solvable in known examples even on complex surfaces with constant right-hand side.
Because only the eigenvector-field derivative estimates use dimension two, the mechanism offers a concrete route toward higher-dimensional Hessian quotients.

Reading between the lines

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

Beyond the stated theorem, the proof's asymptotic structure suggests replacing the affine $\phi(t)=At$ by a strictly convex $\phi$ in (3.1); if that makes the leading term of (3.35) grow with $\lambda_1$, the final contradiction would close with no other changes.
The same concavity identity applies to $\sigma_n/\sigma_l$ for $1\le l<n$, so the test-function mechanism may transfer once higher-dimensional control of the eigenvector-field derivatives is established.
If the bound extends to all closed manifolds, it would show the real equation has no global topological obstruction, so any failure of the higher-dimensional conjectures would originate in the second-order estimate itself rather than in cohomology.

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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

Short version: the new test-function idea is real and the paper is honest about what is already known, but the advertised proof has a load-bearing gap in Proposition 2.4. The problematic step is (2.13), not (3.35). The author commutes u_{11ii} to u_{ii11} and writes an inequality with a −Cλ1 term outside the 1/λ1 factor, then drops it in the next line as if it were bounded by a constant. That is not justified for large λ1; the commutation error should be O(1) after dividing by λ1. Since the structural positive term in (2.10) rests on this, the maximum-principle proof of Theorem 2.2 does not go through as written.

I disagree with the reader on the final step: the objection that the RHS of (3.35) does not grow with λ1 is not the real problem. A uniform positive lower bound is enough there, because the bracket is λ1 + O(1) and F^{11} is roughly F^2/λ1^2, so the leading term is a fixed positive constant. The gap is earlier, in Proposition 2.4.

What the paper does well: Proposition 2.6 is a clean and useful concavity identity. The eigenvector-field computations in Section 3 are explicit, and the test function (3.1) is a genuinely novel idea with clear potential beyond dimension two. The author also deserves credit for Remark 4.1, which states plainly that the two-dimensional theorem is a consequence of Heinz's interior estimate, so the paper does not oversell the result. The real-vs-complex contrast is worth writing about.

Soft spots beyond (2.13): Proposition 2.5 is stated without proof and cited to an in-preparation paper; it is not used in the core argument (Prop 2.6 is), but it is still a citation-practice problem. The passage from interior estimates to the global estimate is handled quickly, and the existence proof in Section 4 inherits all the weight of Theorem 2.2.

Who is this for: anyone working on Hessian quotient equations, a priori estimates, or the Delanoë–Urbas problem. It deserves a serious referee; the right call is major revision, not desk reject. If the (2.13) issue is repairable, the method is interesting enough to matter; if not, the paper reduces to a known theorem with a promising but incomplete new approach.

Referee Report

3 major / 4 minor

Summary. The paper studies the Hessian quotient equation F(u)=σ_n/σ_{n-1}(λ(g^{-1}(χ+∇²u)))=f on closed Riemannian manifolds and proves, in dimension two, an a priori second-order estimate for admissible solutions (Theorem 2.2). The proof introduces a test function involving log λ1 and a penalization by the directional derivative of u along an eigenvector of the largest eigenvalue λ1, and uses a concavity identity for the quotient operator (Proposition 2.6). Theorem 2.3 then derives existence of smooth solutions, up to normalization of the right-hand side, by a standard continuity method. The C⁰ and C¹ estimates are collected in the appendix.

Significance. If correct, Theorem 2.2 would establish that the real positive Hessian quotient equation in dimension two is unobstructed, in contrast to the complex J-equation, and the new maximum-principle mechanism could be a promising route to higher dimensions. The paper is largely self-contained: Proposition 2.6 is an explicit, checkable identity, and the C⁰ and C¹ estimates are proved in the appendix. At the same time, the author candidly notes in Remark 4.1 that the C² estimate itself already follows from Heinz's interior estimate, so the main value lies in the new proof. The current manuscript, however, contains a serious gap in the proof of Proposition 2.4 that is load-bearing for the announced argument.

major comments (3)

[§2.2, Eq. (2.13)] The displayed chain in (2.13) contains the term −Cλ1 on the right-hand side, which is unbounded below as λ1→∞. In the inequality ``≥ (1/λ1)(χ_{ii,11}+u_{ii11}+2Σ...) − Cλ1 + C|∇u|_g + (χ_{11,ii}−χ_{ii,11})/λ1 ≥ (1/λ1)(\tilde g_{ii,11}+2Σ...) − C'', the final ``≥ −C'' is false for large λ1. The commutation of u_{11ii} to u_{ii11} produces an O(λ1) error before division by λ1, hence an O(1) error after division; the manuscript appears to have omitted a factor λ1^{-1} (or to have meant −C rather than −Cλ1). As written, this invalidates the proof of Proposition 2.4 and therefore the lower bound (2.10), which is essential for the maximum-principle step (3.35). This must be corrected and justified explicitly.
[§2.2, Eq. (2.20)] In the passage from the fourth to the fifth displayed line of (2.20), the term −2(F^{ii}\tilde g_{ii,1})²/(Fλ1) is dropped from a chain of lower bounds. Since it is nonpositive, removing it makes the right-hand side larger, and the displayed inequality ``≥'' between those two lines is in the wrong direction. The term can be absorbed using the differentiated equation (2.14), because F^{ii}\tilde g_{ii,1}=f_1 is bounded, but this absorption is not shown. The proof of Proposition 2.4 is therefore incomplete at this point and needs an explicit justification.
[§3.4, Eq. (3.35)] The final lower bound in (3.35) replaces (V²₁u₂ + \tilde g₁₁ − χ₁₁)² by a positive universal fraction of λ₁², but this step is not proved. A quantitative form of the separation assumption (3.20), together with the bounds (3.29) and Proposition 5.2, should imply |V²₁u₂ + \tilde g₁₁ − χ₁₁| ≥ cλ₁ for a constant c>0 independent of λ₁; the author should state this explicitly. I do not think that the alleged issue with the right-hand side of (3.35) being independent of λ₁ is decisive: a uniform positive lower bound would already contradict the maximum principle. The real difficulty is the missing justification of that uniform lower bound.

minor comments (4)

[§2.2, Prop. 2.5] Proposition 2.5 is stated without proof and is cited to reference [18], which is ‚Äúin preparation. Since only Proposition 2.6 is used in the proof of Theorem 2.2, this is not fatal, but it should be clarified whether §2.5 is needed or can be omitted.
[§2.1, Eq. (2.9)] The display (2.9) appears garbled: the middle line seems intended to read something like `1/n ≤ F := Σ_i F^{ii} ≤ 1`, but as typeset it is hard to parse. Please correct the typography.
[§3.4, Eq. (3.20)] The condition ``λ1 >> λ2'' in (3.20) is used quantitatively in (3.28), (3.29), and (3.35). It should be quantified, for example as λ1 ≥ (1+δ)λ2 for a fixed δ>0 when λ1 is large, and the resulting constants should be tracked.
[General notation] The symbol F is used both for the operator value f(u) in (2.3) and for the trace Σ_i F^{ii} in (2.9). In (3.35) this ambiguity could confuse the reader; a different symbol for the sum would help.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the C^2 estimate is derived from an independently proved concavity identity and a new test function; the only self-citation (Proposition 2.5 to an in-preparation paper) is not actually used in the proof.

full rationale

The derivation of Theorem 2.2 is self-contained along its actual chain: the key concavity identity (2.16) for sigma_2/sigma_1 is proved in Proposition 2.6 by direct differentiation of F = 1/sigma_1(lambda^{-1}), and the C^1 bound used in (2.13) and later is proved in Appendix Proposition 5.2. The maximum-principle argument combines the new test function (3.1) with the linearized-operator lower bound (2.10), and no fitted parameter or data-dependent quantity is renamed as a prediction. Proposition 2.5 is stated as a general concavity fact and cited to the in-preparation paper [18] by Guan and Sroka, but the proof of Proposition 2.4 uses only the special-case identity (2.16), so this self-citation is not load-bearing. Remark 4.1 notes that the same estimate can also be obtained from Heinz's interior estimate; that admission reduces the novelty claim but does not make the presented proof circular, since Heinz is not invoked as an input. Concerns about the sign or order of terms in (2.13) and (3.35) are mathematical-estimate issues, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central proof is mostly self-contained, but it pulls in an unproved concavity proposition from an in-preparation self-cited reference and introduces a test-function constant A. No data fitting or invented physical entities appear.

free parameters (1)

A (test function constant phi'=A) = sufficiently large constant
Chosen large relative to C and inf f in (3.35); not fitted to data, but the proof of the claimed bound depends on its existence.

assumptions (5)

ad hoc to paper Proposition 2.5 concavity lower bound (reference [18])
Stated without proof and cited to Guan and Sroka, in preparation; not directly used in the final proof of Theorem 2.2, but presented as a structural ingredient.
standard math Standard elliptic regularity: Schauder and Evans-Krylov estimates
Used in Section 4 to obtain C^{2,alpha} estimates and to bootstrap smoothness.
standard math Curvature commutation formulas for fourth-order derivatives of u
Used throughout Sections 2 and 3, following [2] and [27].
domain assumption Local eigenvector field V exists near the maximum point
Requires lambda1 to be a simple eigenvalue; the paper assumes lambda1 >> lambda2 at the maximum, otherwise lambda2 is bounded.
domain assumption Existence of an admissible v for Theorem 2.3
This is an explicit hypothesis of Theorem 2.3.

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Cite this review

Pith. "Pith review of Remarks on Hessian quotient equations on Riemannian manifolds." pith.science (2026). https://pith.science/paper/A5RQC53G

@misc{pith2026250103386,
  author       = {Pith},
  title        = {Pith review of: Remarks on Hessian quotient equations on Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5RQC53G}},
  note         = {Machine review of arXiv:2501.03386}
}

abstract

We consider Hessian quotient equations in Riemannian setting related to a problem posed by Delano\"e and Urbas. We prove unobstructed second order a priori estimate for the real Hessian quotient equation via the maximum principle argument on Riemannian manifolds in dimension two. This is achieved by introducing new test function and exploiting some fine concavity properties of quotient operator. This result demonstrates that there is intriguing difference between the real case and the complex case, as there are known obstructions for $J$-equation in complex geometry.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Andrews, Contractions of convex hypersurfaces in Euclidian space , Calc

B. Andrews, Contractions of convex hypersurfaces in Euclidian space , Calc. Var., 2, 151–171, 1994

work page 1994

[2] [2]

Brendle, K

S. Brendle, K. Choi, and P. Daskalopoulos, Asymptotic behavior of ﬂows by powers of the Gaussian curvature, Acta Math., 219(1), 1–16, 2017

work page 2017

[3] [3]

Caﬀarelli, L

L. Caﬀarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second–order elliptic equations. III. Functions of the eigenvalues of the Hessian ., Acta Math., 155(3-4), 261–301, 1985

work page 1985

[4] [4]

C. Chen, F. Han, and Q. Ou, The interior C 2 estimate for the Monge-Amp´ ere equation in dimension n = 2, Anal. PDE, 9(6), 1419–1432, 2016

work page 2016

[5] [5]

X. X. Chen, On the lower bound of the Mabuchi energy and its application , Internat. Math. Res. Notices (IMRN), 12, 607–623, 2000

work page 2000

[6] [6]

Cheng andS.-T

S.-Y. Cheng andS.-T. Yau, The real Monge–Amp´ ere equation and aﬃne ﬂat structures , Proceedings of the 1980 Beijing Symposium on Diﬀerential Geometry and Diﬀe rential Equations, Vol. 1, 2, 3 (Beijing, 1980), Science Press Beijing, Beijing, 339–370, 1982

work page 1980

[7] [7]

Delano¨ e, ´Equations du type Monge-Amp` ere sur les vari´ et´ es riemanniennes compactes

P. Delano¨ e, ´Equations du type Monge-Amp` ere sur les vari´ et´ es riemanniennes compactes. I. , J. Func- tional Analysis, 40(3), 358–386, 1981

work page 1981

[8] [8]

Delano¨ e, ´Equations du type de Monge-Amp` ere sur les vari´ et´ es riema nniennes compactes

P. Delano¨ e, ´Equations du type de Monge-Amp` ere sur les vari´ et´ es riema nniennes compactes. II. , J. Functional Analysis, 41(3), 341–353, 1981

work page 1981