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Interior curvature estimate for curvature quotient equations on convex hypersurfaces

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Pith's one-line read For $n\geq 3$, every $C^4$ convex graph solving $\sigma_n/\sigma_{n-2}(\lambda)=f>0$ has all principal curvatures bounded on the inner half-ball; the bound depends only on $n$, $r$, the $C^2$ norm and lower bound of $f$, and the $C^1$…

desk verdict Plausible new interior C^2 estimate for σ_n/σ_{n-2}=f, but the key concavity lemma is deferred to a preprint and private communication — worth refereeing, not yet citable. read the letter →

arxiv 2505.00360 v2 pith:UM3IPVT2 submitted 2025-05-01 math.DG math.AP

Jianxiang Liu This is my paper · ORCID
classification math.DGmath.AP MSC 35J6035B4553C21
keywords curvaturequotientequationinteriorestimateconvexhypersurfaceHessianfullynonlinearellipticJacobiinequalityelementarysymmetricpolynomialsecondfundamentalform
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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

This paper establishes an interior curvature estimate for convex hypersurfaces in $\mathbb{R}^{n+1}$ with $n\geq 3$ that solve the curvature quotient equation $\sigma_n/\sigma_{n-2}(\lambda(X))=f(X)>0$, where $\sigma_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures. The main theorem says that on the inner half-ball $B_{r/2}$ every principal curvature satisfies $|\lambda_i|\leq C$, with $C$ depending only on $n$, $r$, the $C^2$ norm of $f$, its positive lower bound, and the $C^1$ norm of the graph — no boundary data and no boundary curvature information. Interior estimates of this kind are the missing regularity ingredient for compactness and limit arguments in fully nonlinear geometric equations, and they are known to fail for several neighbouring equations, so each positive case delimits where regularity can be expected. The proof is pointwise: it differentiates the equation twice, uses a concavity inequality for $F=\sigma_n/\sigma_{n-2}$ to absorb the second derivatives of curvature through a Jacobi inequality on Riemannian manifolds, and then runs an auxiliary-function maximum argument with a case analysis.

What carries the argument

The load-bearing mechanism is a Jacobi inequality (Lemma 3.1) on Riemannian manifolds: for $b=\ln\lambda_1$, the inequality $\sum_i F^{ii} b_{ii}\geq c(n)\sum_i F^{ii} b_i^2+\sum_i F^{ii} h_{ii}h_{11}-\sum_i F^{ii} h_{ii}^2-C$ holds in the viscosity sense, where $F=\sigma_n/\sigma_{n-2}$ and $h_{ij}$ is the second fundamental form. Its derivation uses the commutator and Codazzi–Gauss identities to reshape second derivatives of curvature, and it rests on a concavity inequality for $F$ (Lemma 2.6), stated with proof deferred to a companion preprint, which is what kills the bad $\sum_i F^{ii} h_{11i}^2$ and $(\sum_i F^{ii}h_{ii1})^2$ terms. The curvature bound itself is finished through the auxiliary function $P=2\log\rho+\log\log\lambda_1-\beta(X,\nu)/(\nu,E_{n+1})+\alpha(\nu,E_{n+1})^{-2}$, maximized at an interior point, together with structural bounds on $F$ and its derivatives in the eigenvalue cone — in particular the reciprocal identity $\sigma_{n-k}/\sigma_{n-l}(\lambda)=\sigma_k/\sigma_l(\lambda^{-1})$, which rewrites $1/F=\sigma_2(\lambda^{-1})$ and identifies the product of the two smallest curvatures as the effective ellipticity scale.

What would settle it

Test inequality (2.8) directly: for $n=3,4,5$ and eigenvalues with large ratio $\lambda_1/\lambda_n$, evaluate the quadratic form in $\xi$ at the vectors $\xi=e_1$ and $\xi_i=1/\lambda_i$; a single violation in the cone $\lambda_1\geq\cdots\geq\lambda_n>0$ would falsify Lemma 2.6 and with it the Jacobi inequality, since every later estimate in Section 3 is derived from (2.8). Alternatively, build a one-parameter family of convex graphs over $B_1$ solving $\sigma_n/\sigma_{n-2}(\lambda)=1$ with bounded $C^1$ norm and compute $\sup |\lambda_i|$ on shrinking interior balls — the theorem predicts the supremum stays bounded, so an observed blow-up at a fixed interior point would contradict the claimed estimate.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.1: for $n\geq 3$, if $M=(x,u(x))$ is a $C^4$ convex graph over a ball $B_r\subset\mathbb{R}^n$ with positive principal curvatures $\lambda=(\lambda_1,\ldots,\lambda_n)$ satisfying $\sigma_n/\sigma_{n-2}(\lambda)=f(X)>0$ with $f\in C^2(B_r)$, then $\sup_{B_{r/2}}|\lambda_i|\leq C$, where $C$ depends only on $n$, $r$, $\|f\|_{C^2(B_r)}$, $\inf_{B_r} f$, and $\|M\|_{C^1(B_r)}$. In other words, the second fundamental form of any such convex solution is uniformly bounded in the interior, with no dependence on the boundary behaviour of the hypersurface. The paper further claims that this is achieved by a pointwise method: a Jacobi-type inequality for $b=\ln\lambda_1$ in the viscosity sense, powered by a concavity inequality for the quotient operator, replaces the integral estimates and Legendre transform used for the corresponding Hessian quotient equation and thereby transports the estimate to the Riemannian (hypersurface) setting.

Load-bearing premise

The proof leans on a single concavity inequality for the operator $\sigma_n/\sigma_{n-2}$, stated as Lemma 2.6, whose proof the paper does not give — it refers to a private communication and a companion preprint — and if that inequality is false or its hypotheses are narrower than assumed, the Jacobi inequality and the curvature bound collapse.

Editorial extensions

If this is right

  • Every $C^4$ locally convex graph over a ball solving $\sigma_n/\sigma_{n-2}(\lambda)=f>0$ has all principal curvatures bounded on $B_{r/2}$ by a constant depending only on $n$, $r$, $\|f\|_{C^2}$, $\inf f$, and $\|M\|_{C^1}$, so interior curvature concentration cannot occur while those data stay under control.
  • Because the bound is independent of boundary values, it supplies the interior regularity step needed to run compactness and limit arguments for solutions of this curvature quotient equation.
  • The pointwise Jacobi-inequality method applies directly on hypersurfaces, avoiding the Legendre transform and integral techniques used for the Euclidean Hessian quotient analogue, and thereby gives a first interior curvature bound of this kind for the gap-two quotient $\sigma_n/\sigma_{n-2}$ in the curvature setting.
  • The inequalities are proven in the viscosity sense, so the curvature bound is stable under $C^2$ approximation and passes to limits of convex solutions.

Reading between the lines

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

  • The same auxiliary function and Jacobi inequality should extend to right-hand sides $f(X,\nu(X))$ with comparable $C^2$ control and a positive lower bound, since the proof never uses the special form $f(X)$ beyond the differentiation step and norm dependence.
  • If a gap-two analogue of the concavity inequality (2.8) holds for $\sigma_k/\sigma_{k-2}$ with $k<n$, the machinery here would plausibly settle the still-open interior estimate for the Hessian quotient equation in the Euclidean setting, because the reciprocal identity reduces the structure to a $\sigma_2$-type inverse spectrum.
  • The paper's engine, inequality (2.8), is stated without proof, attributed to a private communication, and deferred to a companion preprint; an independent check of (2.8) on the cone $\lambda_1\geq\cdots\geq\lambda_n>0$ is the single most valuable verification, since every subsequent step in Section 3 derives from it.
  • The acknowledgements state that this version corrects errors in the author's thesis manuscript, and several displays in Section 4 still contain typographical slips; the corrected version should be read as authoritative when checking the case analysis.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims an interior curvature estimate for convex graphs M=(x,u(x)) over a ball B_r in R^n satisfying the curvature quotient equation sigma_n/sigma_{n-2}(lambda)=f(X)>0 with n>=3. The main result, Theorem 1.1, asserts that sup_{B_{r/2}} |lambda_i| is bounded by a constant depending only on n, r, ||f||_{C^2(B_r)}, inf_{B_r} f, and ||M||_{C^1(B_r)}. The proof develops a pointwise Jacobi inequality (Lemma 3.1) using a concavity inequality for F=sigma_n/sigma_{n-2} (Lemma 2.6), then applies the Guan-Qiu auxiliary function P(X)=2 log rho + log log lambda_1 - beta (X,nu)/(nu,E_{n+1}) + alpha/(nu,E_{n+1})^2, followed by a case-by-case analysis of the terms involving rho and the horizontal coordinates. The author states that the method is inspired by Lu's work on Hessian quotient equations and avoids the Legendre transform, which is the main novelty relative to the prior Hessian-quotient results.

Significance. If the proof is correct, this would be a meaningful advance: it provides interior curvature estimates for the curvature quotient equation sigma_n/sigma_{n-2}=f in all dimensions n>=3, a case that the paper correctly identifies as open. The pointwise Jacobi inequality approach is a genuine methodological difference from Lu's integral method, and the auxiliary-function part is clear in organization. The paper also gives explicit structural lemmas for the operator F, including the concavity inequality (2.8), and the proof strategy is transparent. However, the significance is heavily conditional: the central concavity inequality is not proved in the manuscript, and there are algebraic identities in Section 2 that appear to be incorrect as written. Because these points support the main estimate, the contribution cannot be fully assessed in its current form.

major comments (3)
  1. [Section 2, Lemma 2.4, Eq. (2.6)] The identity sum_i F_ii h_ii = F is not correct as stated. Since F = sigma_n/sigma_{n-2} is homogeneous of degree 2 in the eigenvalues lambda_i, Euler's theorem gives sum_i F_ii lambda_i = 2F, not F. This identity is used in the proof of Theorem 1.1 in Section 4, specifically in Eq. (4.11) where the term 4 F_ii h_ii[(X,nu)-(X,E_{n+1})(nu,E_{n+1})]/rho is replaced by -4F[(X,nu)-(X,E_{n+1})(nu,E_{n+1})]/rho based on (2.6). If the correct homogeneity factor is 2F, the numerical factor changes and the subsequent constants must be recalculated. This is a load-bearing algebraic step, and the error suggests that the identities in Section 2 have not been checked carefully.
  2. [Section 2, Lemma 2.6; Section 3, Eq. (3.15)] Lemma 2.6 is the essential concavity inequality for F=sigma_n/sigma_{n-2}, stated as inequality (2.8). It is the only tool that eliminates the second-derivative-of-curvature terms in the proof of Lemma 3.1: after combining the differentiated equation with the commutator identity, the cross terms are bounded using (2.8) at the step following Eq. (3.15). The manuscript attributes this inequality to a private communication [7] and defers a proof to the preprint [13]. As written, Theorem 1.1 is therefore conditional on an externally stated, unverified lemma. The author should include a complete proof of (2.8) in the manuscript, or at minimum state it as a theorem with a precise reference to a published or publicly verifiable source, and confirm that the hypotheses cover the application with xi_i=h_ii1 at a point where h is diagonalized. Without this, the Jacobi inequality and the main estimate are not established within the paper.
  3. [Section 4, transition from Eq. (4.39) to Eq. (4.40)] The passage from inequality (4.39) to inequality (4.40) contains a loss of a factor of h_11. In (4.39) the positive term is F_nn h_11n^2 / (20 h_11^2 log h_11), while (4.40) states F_nn h_11n^2 / (20 h_11 log h_11). Additionally, the coefficient estimate in (4.39) is written as d/(lambda_n rho) - |...|, but (4.40) uses d/(2 lambda_n rho) in the final lower bound; the justification for replacing d by d/2 is not given, and the nonnegativity claim requires a quantitative lower bound on lambda_1 that is only heuristically described as 'lambda_1 is sufficiently large'. This step carries the estimate in Case 3.2 and must be rewritten with consistent factors and explicit inequalities.
minor comments (4)
  1. [Section 2, Lemma 2.4, proof of (2.4)] In inequality (2.4), the lower bound for F_nn is written as F^2/(lambda_i^2 lambda_{n-1}), but the intended expression should have lambda_n^2 in the denominator, since F_nn = F^2/lambda_n^2 * sum_{k != n} 1/lambda_k. The later use in Case 3.2 indeed uses lambda_n^2, so this appears to be a typo in the lemma statement.
  2. [Section 2, Lemma 2.4, proof of (2.7)] In the proof of (2.7), the displayed inequalities conclude with C_3(n)F/lambda_n and C_2(n)F/lambda_n, whereas the lemma statement has F^2/lambda_n on both sides. Since F_ii h_ii^2 = F^2 sum_{k!=i} 1/lambda_k, the square is correct in the statement; the proof is missing the factor F in the displayed estimates.
  3. [Section 4, Eq. (4.17) and discussion after it] The sentence beginning 'Since we can choose h_11, alpha to be large enough, and applying formula (3.9)' appears before the displayed inequality that already includes the result of that choice. The logical order should be clarified: formula (3.9) is applied to the term containing (lambda_1)_i^2, but the display (4.17) is written after dropping the negative terms, which should be explained explicitly.
  4. [References] References [7] and [8] are listed as private communications; for a published journal, such sources cannot support the main technical lemma. Reference [13] should be cited with a theorem statement or a specific equation number so that the reader can verify the quoted result.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the estimate is not fitted or assumed; reliance on Lemma 2.6 from [13]/[7] is external, though unverified.

full rationale

I walked the derivation chain from the definition F = sigma_n/sigma_{n-2} through the Jacobi inequality (Lemma 3.1) to the auxiliary-function maximum argument in Section 4. The target quantity is the interior bound on the largest principal curvature; it is not assumed at any point, and no fitted parameter is renamed as a prediction. The proof instead differentiates the equation, applies the structure of F, and uses a maximum principle with an auxiliary function, which is a standard a priori estimate pattern. The only load-bearing imported ingredient is Lemma 2.6, the concavity inequality (2.8), which the paper attributes to private communication [7] and to a proof in [13]. This is external support from other authors, not a self-citation chain, and it is not derived from the theorem being proved, so it does not make the argument circular. There is a separate correctness concern noted by a skeptical reader: equation (2.6), sum_i F_ii h_ii = F, appears inconsistent with Euler's identity for the degree-2 homogeneous function F, which would give sum_i F_ii h_ii = 2F. That is a mathematical-error concern, not a circularity concern, and I do not score it under circularity per the instructions. Also, the fact that Lemma 2.6 is unverified in this manuscript weakens certainty in the proof but does not reduce the conclusion to its own inputs. Overall, I found no circular step, so the appropriate score is 1 rather than 0 only because the central inequality is sourced to an unpublished private communication and a separate preprint, making the proof conditional on external verification.

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

No free parameters are fitted to data; the α and β in the auxiliary function are large constants chosen in the proof, not empirical inputs. The main external inputs are algebraic lemmas from prior work and one private communication, listed as axioms above.

assumptions (4)
  • domain assumption Lemma 2.6 concavity inequality (2.8) for F=σ_n/σ_{n-2}
    Stated in Section 2 with proof deferred to preprint [13] and private communication [7]; it is the key algebraic input for the Jacobi inequality in Lemma 3.1.
  • standard math Eigenvalue formulas (3.8)-(3.9) from Lemma 5 of [1] in viscosity sense at repeated eigenvalues
    Used in Lemma 3.1 to differentiate the largest principal curvature at points of multiplicity m.
  • standard math Known Hessian quotient identities in Lemma 2.2 and Lemma 2.3, with proofs in [13]
    Used throughout Sections 2-4 to estimate F_ii, F^{ii,jj}, and eigenvalue bounds.
  • domain assumption The graph has (ν,E_{n+1}) bounded above and below for x in B_1
    Used in the auxiliary function P and in the Case 3 analysis; valid for C^1 graphs over a bounded ball.

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Cite this review

Pith. "Pith review of Interior curvature estimate for curvature quotient equations on convex hypersurfaces." pith.science (2026). https://pith.science/paper/UM3IPVT2

@misc{pith2026250500360,
  author       = {Pith},
  title        = {Pith review of: Interior curvature estimate for curvature quotient equations on convex hypersurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UM3IPVT2}},
  note         = {Machine review of arXiv:2505.00360}
}
abstract

We study interior curvature estimates for convex graphs which satisfy the quotient equation $\frac{\sigma_{n}}{\sigma_{n-2}}(\lambda)=f(X)>0$ in this paper.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 22 canonical work pages

  1. [7]

    Pengfei Guan and Marcin Sroka, Private communication

    work page

  2. [13]

    , Interior C2 estimate for hessian quotient equation in general dimensio n, arXiv preprint arXiv:2401.12229 (2024), https://arxiv.org/abs/2401.12229

    arXiv 2024

  3. [1]

    219 (2017), no

    Simon Brendle, Kyeongsu Choi, and Panagiota Daskalopoulos, Asymptotic behavior of flows by powers of the gaussian curvature , Acta Math. 219 (2017), no. 1, 1–16. MR 3666840

    work page 2017

  4. [2]

    Caffarelli, Louis Nirenberg, and Joel Spruck, The Dirichlet problem for nonlinear second order elliptic equations

    Luis A. Caffarelli, Louis Nirenberg, and Joel Spruck, The Dirichlet problem for nonlinear second order elliptic equations. III. functions of the eigenvalues of th e Hessian , Acta Math. 155 (1985), 261–301. MR 782625 [3] , Nonlinear second-order elliptic equations. V. the Dirichl et problem for Weingarten hypersur- faces, Comm. Pure Appl. Math. 41 (1988), ...

    work page 1985

  5. [4]

    PDE 9 (2016), no

    Chuanqiang Chen, Fei Han, and Qianzhong Ou, The interior C2 estimate for the Monge–Amp` ere equa- tion in dimension n = 2, Anal. PDE 9 (2016), no. 6, 1419–1432. MR 3562353

    work page 2016

  6. [5]

    Pure Appl

    Jingyi Chen, Micah Warren, and Yu Yuan, A priori estimate for convex solutions to special Lagrangia n equations and its application , Comm. Pure Appl. Math. 62 (2009), no. 4, 583–595. MR 2481743

    work page 2009

  7. [6]

    Pengfei Guan and Guohuan Qiu, Interior C2 regularity of convex solutions to prescribing scalar curva ture equations, Duke Math. J. 168 (2019), no. 9, 1641–1663. MR 3959824

    work page 2019

  8. [8]

    Pengfei Guan and Xiangwen Zhang, Private communication. 16

    work page

Show all 24 references
  1. [9]

    Erhard Heinz, On elliptic Monge–Amp` ere equations and Weyl’s embedding p roblem, J. Anal. Math. 7 (1959), 1–52. MR 0100082

    1959

  2. [10]

    Trudinger, The Dirichlet problem for the prescribed curvature quotien t equations , Topol

    Mi Lin and Neil S. Trudinger, The Dirichlet problem for the prescribed curvature quotien t equations , Topol. Methods Nonlinear Anal. 3 (1994), no. 2, 307–323. MR 1301333

    1994

  3. [11]

    Jiakun Liu, Interior C2 estimates for the Monge–Amp` ere equation in dimension two , Proc. Amer. Math. Soc. 149 (2021), no. 6, 2479–2486. MR 4246799

    2021

  4. [12]

    Siyuan Lu, Interior C2 estimate for hessian quotient equation in dimension three , arXiv preprint arXiv:2311.05835 (2023), https://arxiv.org/abs/2311.05835

    2023 arXiv

  5. [14]

    Matt McGonagle, Chong Song, and Yu Yuan, Hessian estimates for convex solutions to quadratic Hessian equation , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire36 (2019), no. 2, 451–454. MR 3913193

    2019

  6. [15]

    Connor Mooney, Strict 2-convexity of convex solutions to the quadratic Hessian equ ation, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2473–2477. MR 4246799

    2021

  7. [16]

    Aleksei Vasil’evich Pogorelov, The Minkowski multidimensional problem , Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C., 1978. MR 0506371

    1978

  8. [17]

    Guohuan Qiu, Interior Hessian estimates for sigma 2 equations in dimension three , Front. Math. China 13 (2018), no. 4, 901–915. MR 3869703

    2018

  9. [18]

    , Interior curvature estimates for hypersurfaces of prescri bing scalar curvature in dimension three, Amer. J. Math. 146 (2024), no. 1, 1–34. MR 4567890

    2024

  10. [19]

    Ravi Shankar and Yu Yuan, Hessian estimate for semiconvex solutions to the Sigma-2 eq uation, Calc. Var. Partial Differential Equations 59 (2020), no. 1, 1–12. MR 4055992

    2020

  11. [20]

    , Hessian estimates for the Sigma-2 equation in dimension fou r, Ann. of Math. (2) 201 (2025), no. 2, 489–513. MR 4880431

    2025

  12. [21]

    Weimin Sheng, John Urbas, and Xu-Jia Wang, Interior curvature bounds for a class of curvature equa- tions, Duke Math. J. 123 (2004), no. 2, 235–264. MR 2066943

    2004

  13. [22]

    John I. E. Urbas, On the existence of nonclassical solutions for two classes o f fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), no. 2, 355–382. MR 1078740

    1990

  14. [23]

    Dake Wang and Yu Yuan, Hessian estimates for special Lagrangian equations with cr itical and super- critical phases in general dimensions , Amer. J. Math. 136 (2014), no. 2, 481–499. MR 3194490

    2014

  15. [24]

    Pure Appl

    Micah Warren and Yu Yuan, Hessian estimates for the Sigma-2 equation in dimension thr ee, Commun. Pure Appl. Math. 62 (2009), no. 3, 305–321. MR 2483569

    2009

  16. [25]

    Xingchen Zhou, Notes on generalized special Lagrangian equation , Calc. Var. Partial Differential Equa- tions 63 (2024), no. 8, 197. Department of Mathematics & Statistics, McMaster Universi ty Email address : liu2238@mcmaster.ca 17

    2024

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