{"id":"f6995460-dc6a-4866-be7d-7ce3b0b56510","arxiv_id":"2505.00360","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves an interior bound on principal curvatures for convex graphs solving the curvature quotient equation σ_n/σ_{n-2}=f in all dimensions n≥3.","lead":"The paper proves that convex surfaces obeying a special curvature equation cannot bend too sharply in their interior. It is a regularity estimate that supports existence and convergence arguments for a class of geometric partial differential equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 hinges on the unproved concavity inequality (2.8); until Lemma 2.6 is verified, the Jacobi inequality and the main estimate are not established.","rationale":"The paper presents a coherent and plausible strategy: a pointwise Jacobi inequality followed by a Guan-Qiu type auxiliary function argument. If Lemma 2.6 were available and correct, the main estimate would likely follow, and I found no internal contradiction that shows Theorem 1.1 is false. However, the proof of Theorem 1.1 flows through Lemma 3.1, and Lemma 3.1 flows through Lemma 2.6, which is neither proved nor even fully stated with a derivation in this manuscript. The manuscript explicitly sends the reader to a private communication and to another preprint for the central concavity inequality. The Euler homogeneity error in Lemma 2.4(2.6) strengthens this concern: sum_i F_ii h_ii is claimed to equal F, but for the degree-2 homogeneous function F the correct identity is 2F. This is a real algebraic mistake in the submitted text, and it indicates that the surrounding identities have not been verified line by line. The remaining slips in denominators and constants appear absorbable into the generic constants C, so they do not by themselves overturn the argument. Because the missing lemma and the Euler error concern exactly the mechanism that controls derivative-of-curvature terms, the paper should not be accepted as fully verified. Since the issue is likely fixable if Lu's preprint is correct, CONDITIONAL remains the right verdict.","tokens_in":15806,"tokens_out":26752,"duration_ms":247339,"concrete_test":"Retrieve the proof of Lemma 2.6 from Lu's preprint [13] (arXiv:2401.12229) and check whether (2.8) is proved there in the stated form. Specifically, verify the proof against the correct Euler identity sum_i F_ii lambda_i = 2F, not F as stated in (2.6), and confirm that the inequality holds for all xi in R^n and all lambda_1 >= ... >= lambda_n > 0. Cross-check with the explicit formula F_ii = F^2 / lambda_i^2 * sum_{k != i} 1/lambda_k. If the proof in [13] is absent or depends on the wrong homogeneity, Theorem 1.1 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's Jacobi inequality (Lemma 3.1) is the engine of the proof, and its derivation applies Lemma 2.6 at inequality (3.15) to eliminate the second derivatives of h. Lemma 2.6 is stated without proof: the manuscript says it was introduced in [7] first and that a proof can be found in [13], where [7] is a private communication and [13] is a separate preprint on Hessian quotient equations. If (2.8) is false, or holds only under extra hypotheses not stated here, the cross terms in Lemma 3.1 cannot be controlled and Theorem 1.1 collapses. The concern is concrete: Lemma 2.4(2.6) asserts sum_i F_ii h_ii = F, while Euler's identity for the degree-2 homogeneous F = sigma_n / sigma_{n-2} gives sum_i F_ii h_ii = 2F. This shows the algebraic identities in Section 2 have not been checked carefully, and the same unchecked algebra underlies the concavity inequality imported from [13]. Thus the central claim is, as written, conditional on an external, unverified lemma.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16015,"tokens_out":5717,"duration_ms":54011,"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":[{"comment":"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.","section":"Section 2, Lemma 2.4, Eq. (2.6)"},{"comment":"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.","section":"Section 2, Lemma 2.6; Section 3, Eq. (3.15)"},{"comment":"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.","section":"Section 4, transition from Eq. (4.39) to Eq. (4.40)"}],"minor_comments":[{"comment":"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.","section":"Section 2, Lemma 2.4, proof of (2.4)"},{"comment":"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.","section":"Section 2, Lemma 2.4, proof of (2.7)"},{"comment":"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.","section":"Section 4, Eq. (4.17) and discussion after it"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main result is structurally interesting and the pointwise method is a reasonable approach, but the manuscript relies on a central inequality that is not proved and whose external reference is a private communication and an arXiv preprint. The Euler homogeneity error in Lemma 2.4(2.6) and the factor loss between (4.39) and (4.40) indicate that the algebraic details have not been fully checked. I recommend major revision with the requirement that the proof of Lemma 2.6 be included or precisely referenced, and that all homogeneity factors be corrected and propagated through Section 4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper claims an interior C^2 estimate for the curvature quotient equation σ_n/σ_{n-2}(λ)=f on convex graphs in R^{n+1}, for all n≥3. That specific equation is indeed not covered by the previous literature — Sheng-Urbas-Wang and Guan-Zhang treated σ_k/σ_{k-1}, Lu treated Hessian quotients. So if the proof is right, Theorem 1.1 is a genuine new result. The strategy is a pointwise Jacobi inequality plus the Guan-Qiu auxiliary function, and the case analysis in Section 4 is carried out with some care. That is real work, and the author clearly knows the surrounding papers.\n\nThe problem is that the paper is not self-contained at the one place that matters. Lemma 2.6, the concavity inequality that controls the cross terms in the Jacobi inequality, is stated without proof, attributed to a private communication [7] and a preprint [13]. The entire estimate collapses without it. A referee would need the proof, or at least a precise statement with all hypotheses, before taking the theorem seriously. The fact that [13] is on arXiv helps, but it does not replace having the lemma proved in the paper.\n\nThere are also small algebraic slips that suggest Section 2 was not checked carefully. Equation (2.6) says Σ F_ii h_ii = F; by Euler's identity for a degree-2 homogeneous F, it should be 2F. The bounds used later absorb that factor into constants, so it is probably harmless, but it is still wrong as displayed. The denominator change between (4.39) and (4.40) is justified by (2.2) but not spelled out. These are fixable, but they dent confidence in the algebra.\n\nThe central claim is plausible; I do not see circular dependence or fitting-to-data problems. The main burden is the missing proof of Lemma 2.6.\n\nRecommendation: send it to a serious referee, but insist that the proof of Lemma 2.6 be included or that the author state exactly where it is proved and under which hypotheses. If that lemma is true, the result is worth having. The author should also fix the homogeneity identity and the denominator step before resubmission.","headline":"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.","tokens_in":16592,"tokens_out":6034,"would_cite":false,"duration_ms":52125,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35B45","53C21"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $C^1$…","keywords":["curvature quotient equation","interior curvature estimate","convex hypersurface","Hessian quotient equation","fully nonlinear elliptic equation","Jacobi inequality","elementary symmetric polynomial","second fundamental form"],"falsifier":"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.","tokens_in":15568,"feed_emoji":"📐","tokens_out":21184,"duration_ms":175575,"temperature":0.7,"pith_summary":"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.","feed_headline":"Curvatures of convex graphs cannot blow up inside the ball","feed_subtitle":"Every solution of the quotient equation σ_n/σ_{n−2}=f gets a uniform half-ball bound — no boundary data needed.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Companion preprint whose Jacobi inequality for the Hessian quotient equation is generalized here to Riemannian manifolds, and which is cited as the source of the proof of the concavity inequality (2.8).","marker":"[13]"},{"why":"Private communication credited as the original source of the concavity inequality (2.8) that powers the Jacobi inequality.","marker":"[7]"},{"why":"Supplies the auxiliary function whose maximum-point computation yields the final curvature bound.","marker":"[6]"},{"why":"Provides the viscosity-sense derivative formulas for the largest eigenvalue that start the Jacobi-inequality derivation.","marker":"[1]"},{"why":"Source of the elementary symmetric polynomial identities and Hessian-operator formulas used in the preliminaries.","marker":"[10]"},{"why":"Foundational interior estimate for the quadratic Hessian equation in dimension three, the template for pointwise interior estimates of this type.","marker":"[24]"},{"why":"Counterexample results showing where interior estimates fail, recalled to frame why the gap-two quotient is the meaningful boundary case.","marker":"[22]"}],"fun_headline_variants":["Interior curvature bound for convex quotient graphs","No curvature blow-up inside convex hypersurfaces","Uniform curvature control in half-ball interior","Convex graphs: curvature stays bounded inside","Quotient equation yields interior curvature estimate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Interior curvature bound for convex quotient graphs","No curvature blow-up inside convex hypersurfaces","Uniform curvature control in half-ball interior","Convex graphs: curvature stays bounded inside","Quotient equation yields interior curvature estimate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000712,"raw_usage":{"total_tokens":3142,"prompt_tokens":825,"completion_tokens":2317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":2251}},"tokens_in":441,"tokens_out":2317,"duration_ms":18994,"temperature":1.0,"reasoning_tokens":2251,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:46:11.342482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Private communication credited as the original source of the concavity inequality (2.8) that powers the Jacobi inequality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the auxiliary function whose maximum-point computation yields the final curvature bound."},{"cited_title":"219 (2017), no","cited_arxiv_id":null,"evidence_quote":"Provides the viscosity-sense derivative formulas for the largest eigenvalue that start the Jacobi-inequality derivation."},{"cited_title":"Trudinger, The Dirichlet problem for the prescribed curvature quotien t equations , Topol","cited_arxiv_id":null,"evidence_quote":"Source of the elementary symmetric polynomial identities and Hessian-operator formulas used in the preliminaries."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Foundational interior estimate for the quadratic Hessian equation in dimension three, the template for pointwise interior estimates of this type."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Counterexample results showing where interior estimates fail, recalled to frame why the gap-two quotient is the meaningful boundary case."}],"review_version":1}