{"id":"ad328263-091e-4b53-a145-011113431eef","arxiv_id":"2501.03386","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A dimension-two a priori second-order estimate for Hessian quotient equations is claimed via a new test function, but the final contradiction does not actually constrain the largest Hessian eigenvalue.","lead":"This paper gives a maximum-principle proof attempt for a second-order a priori estimate for the real Hessian quotient equation on closed Riemannian surfaces, and derives an existence theorem. The proof introduces a new test function involving the directional derivative of the solution along an eigenvector field, but the final maximum-principle contradiction appears not to depend on the quantity it is meant to bound.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The genuine gap is in Proposition 2.4: inequality (2.13) contains an unbounded -Cλ1 term that makes the stated estimate false; the reader's objection to (3.35) is not the real issue.","rationale":"The reader's weakest_assumption targets the final inequality (3.35) and claims the RHS must grow with λ1. This is a misreading: the maximum-principle contradiction only needs a uniformly positive lower bound. At a maximum of Q, the bracket (V_1^2 u2 + tilde g_11 − χ_11) equals λ1 + O(1) because tilde g_11 = λ1 + χ_11 and V_1^2 u2 is bounded by (3.29), while F^11 = F²/λ1²; hence the first term in (3.35) is of order F² φ'² u1², and even the F² φ'/2 term alone gives a positive RHS for sufficiently large constant φ' = A. Thus the proof strategy is not flawed for the reason given by the reader. However, a careful check of Proposition 2.4 reveals a more serious written error: inequality (2.13) contains a term −Cλ1 that is unbounded below as λ1→∞, making the displayed chain of inequalities false. Since Proposition 2.4 supplies the crucial F^11 tilde g_11,1²/λ1² term in the final maximum-principle inequality, the proof of Theorem 2.2 is not valid as written. The error appears to be a typo—the correct bound should be −C after dividing by λ1—but as stated, it breaks the argument. This is a load-bearing concern independent of the reader's objection, and it supports rejecting the paper's advertised new proof, although the theorem may still be true by the interior estimate cited in Remark 4.1. I therefore keep the REJECT verdict, but for a different reason than the reader's.","tokens_in":14783,"tokens_out":30167,"duration_ms":241730,"concrete_test":"Re-derive the commutator bound for u11ii − uii11 in normal coordinates (2.7) using the formula (2.8). The curvature terms contain R * u_ab, with a,b possibly (1,1), so u11ii − uii11 = O(λ1) + O(1); after division by λ1 the error is O(1). Verify the passage in (2.13): if the second inequality is corrected to −C + (C|∇u|)/λ1, then Proposition 2.4 holds; if the −Cλ1 is kept, (2.10) fails and the proof of Theorem 2.2 collapses.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing gap is in the proof of Proposition 2.4, not in the maximum-principle step (3.35). In (2.13) the author asserts (1/λ1)(χ11,ii + u11ii) ≥ (1/λ1)(χii,11 + uii11) − Cλ1 + C|∇u| + (χ11,ii − χii,11)/λ1 ≥ (1/λ1)(tilde g_ii,11) − C. The middle term −Cλ1 is unbounded below as λ1→∞, so the second inequality is false. The commutation formula u11ii = uii11 + R * (∇²u) + lower-order terms gives an O(λ1) error before division by λ1, hence O(1) after division; the correct lower bound is −C, not −Cλ1. Since (2.10) relies on (2.13), the positive term F^11 tilde g_11,1²/λ1² used in (3.35) is not justified as written. The reader's concern that the RHS of (3.35) does not grow with λ1 is not valid: a uniform positive lower bound suffices for a maximum-principle contradiction, because the bracket is λ1 + O(1) and F^11 ~ F²/λ1². The genuine issue is the −Cλ1 term, which would make the RHS negative for large λ1 and destroy the contradiction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15079,"tokens_out":23191,"duration_ms":194926,"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":[{"comment":"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.","section":"§2.2, Eq. (2.13)"},{"comment":"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.","section":"§2.2, Eq. (2.20)"},{"comment":"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.","section":"§3.4, Eq. (3.35)"}],"minor_comments":[{"comment":"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.","section":"§2.2, Prop. 2.5"},{"comment":"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.","section":"§2.1, Eq. (2.9)"},{"comment":"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.","section":"§3.4, Eq. (3.20)"},{"comment":"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.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and contains a promising new maximum-principle idea, but the proof of Proposition 2.4 has at least one load-bearing gap that must be repaired. If the missing arguments are supplied, the paper could be publishable as a new proof of a known-type estimate in dimension two. Please also ask the author to remove or fully prove the dependence on the in-preparation reference [18]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nI 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.\n\nWhat 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.\n\nSoft 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.\n\nWho 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.","headline":"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.","tokens_in":15617,"tokens_out":5994,"would_cite":false,"duration_ms":50334,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J05","35R01"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new maximum-principle test function yields unobstructed second-order estimates for the real Hessian quotient equation in dimension two.","keywords":["Hessian quotient equations","fully nonlinear elliptic equations","second-order a priori estimates","maximum principle","Riemannian manifolds","positive Hessian quotient operators","J-equation","Monge–Ampère equation"],"falsifier":"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.","tokens_in":14538,"feed_emoji":"📐","tokens_out":14065,"duration_ms":123610,"temperature":0.7,"pith_summary":"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.","feed_headline":"No obstruction blocks Hessian quotient estimates in dimension two","feed_subtitle":"A new test function plus a concavity identity prove $\\lvert\\nabla^2 u\\rvert\\le C$, where the complex J-equation hits genuine obstructions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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