{"id":"8da99e26-cc2f-4889-b920-8d22c4dd84e9","arxiv_id":"2506.09639","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A necessary and sufficient solvability criterion is extended to almost Hermitian manifolds with gradient terms, yielding explicit infimum formulas for the twisted constants in Calabi-Yau type equations.","lead":"This paper proves a solvability criterion for a general class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, extending a recent result by Guo and Song. It then uses the criterion to give explicit infimum formulas for the twisted constant in several Calabi-Yau type equations, answering a question posed by Chu, Tosatti and Weinkove.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Closedness of the continuity method rests on unverified imports of [10, Prop. 3.11] and [9, Prop. 3.2]; until their hypotheses are checked for family (3.3), Theorem 2.5 and the twisted-constant formulas are not established.","rationale":"The central claim is the infimum formula for the twisted constant. It is obtained by applying Theorem 2.5 to the form-type and Hessian reductions; those reductions are sound (T is invertible, f=(S_n(T))^{1/n} satisfies (a)-(d), Γ=T^{-1}(Γ_n) contains the positive cone, and the P_χ identity is correct). The openness part of the continuity method is standard and the elliptic/maximum-principle arguments there are plausible. The closedness part is not self-contained: uniform estimates are quoted from [10] and the C-subsolution construction is quoted from [9]. This is a genuine proof gap in the write-up, not a disagreement with consensus; it is fixable by supplying the missing verifications. Minor issues (the garbled statement of Theorem 2.5, the 'equation (1.6)' typo, and the unproved positivity of σ) do not change the conditional verdict.","tokens_in":7960,"tokens_out":17164,"duration_ms":189897,"concrete_test":"Verify [10, Proposition 3.11] hypotheses for (3.3) in the concrete case f=(S_n(T))^{1/n}, Γ=T^{-1}(Γ_n), Z≠0, n=2; compute the C-subsolution constants r,R for u and compare with [10]'s definition. In particular, confirm that ψ_t+c_t has the regularity and positivity required and that the estimate constants depend only on sup_t ||ψ_t+c_t||_{C^0}, r, R, and the manifold. Also re-derive Lemma 3.2 with the Z and [e_i,e_j] terms written out; if the proof in [9] uses an integration by parts or a comparison argument that fails for these first-order perturbations, the bound -C≤c_t≤tc is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 closes the continuity method in Lemma 3.11 by invoking Huang-Zhang [10, Prop. 3.11] for the family (3.3), while Proposition 3.8, which is supposed to produce the required uniform C-subsolution, is dismissed with 'See [9, Proposition 3.2]'. Neither import is verified in the present setting: (3.3) contains gradient terms Z(∂u), lives on a non-integrable almost complex manifold, and has a t-dependent right-hand side ψ_t+c_t. If [10, Prop. 3.11] has hypotheses (fixed RHS, fixed cone/operator, or a C-subsolution in a slightly different sense) that the family (3.3) does not satisfy, the closedness of T fails and Theorem 2.5 is not proved; Theorem 1.3 depends on it directly. The author should either prove Lemma 3.2, Proposition 3.6, and Proposition 3.8 in this generality, or spell out the exact hypotheses of [10, Prop. 3.11] and verify each one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fully nonlinear elliptic equations F[u] = e^ψ on compact almost Hermitian manifolds, where F[u] = f(λ(ω_u)) and ω_u = ω + √−1∂∂̄u + Z(∂u) includes a linear gradient term. It states a Guo-Song type criterion (Theorem 2.5) that identifies solvability with the existence of a sub-solution in the sup-slope sense, and it uses this criterion to derive explicit inf-sup formulas for the twisted constant c in two Calabi-Yau type equations: form-type equations with gradient terms (Theorem 1.3, case (1)) and Hessian type equations (Theorem 1.3, case (2)). The paper aims to address a question raised by Chu-Tosatti-Weinkove.","tokens_in":8255,"tokens_out":8944,"duration_ms":95598,"significance":"If the stated results are established, the paper provides a clean variational characterization of the twisted constant and directly extends recent work of Guo-Song to the almost Hermitian, gradient-term setting. The final formulas in Theorem 1.3 are explicit and concrete, and the overall strategy—reducing Theorem 1.3 to the criterion and then using a continuity method with a normalized constant—is sensible. However, the proof is not self-contained in its current form: several load-bearing lemmas are quoted from Guo-Song [9] and Huang-Zhang [10] without checking that their hypotheses hold for the deformed family (3.3) with gradient terms on a non-integrable almost complex manifold. The central claims therefore remain conditional on these unverified imports.","major_comments":[{"comment":"The theorem as stated is internally inconsistent and does not match the proof that follows. Condition (1) says that a solution to (1.1), i.e. F[u] = e^ψ, is equivalently e^{-ψ}F[u] = constant, but for a solution of (1.1) this constant is identically 1. The proof in Section 3 (Corollary 3.12 and Proposition 3.9) actually constructs a solution to F[u] = σe^ψ. Moreover, the printed conditions (3) and (4) are indistinguishable: both read max_M e^{-ψ}F[u] < min_M e^{-ψ}F∞[u], although the proof requires a pair (underline u, bar u) satisfying inequality (3.1). The theorem must be restated precisely, for example in terms of solvability of F[u] = e^{ψ+c} for some constant c = log σ, and the super-solution/sub-solution notation must be made explicit.","section":"Section 2, Theorem 2.5"},{"comment":"Lemmas 3.2, 3.3, 3.5 and Proposition 3.6 are introduced with the sentence 'Proofs of following lemmas can be found in [9], hence we omit them.' The manuscript's setting is more general than that of Guo-Song [9], because the operator contains the gradient term Z(∂u) and the manifold is almost Hermitian rather than integrable Hermitian. These lemmas are load-bearing: Lemma 3.2 gives uniform bounds on c_t, Lemma 3.3 controls f∞, Proposition 3.6 produces the lower bound used to construct sub-solutions, and Proposition 3.8 is the basis for the a priori estimates. The paper should either prove these statements in the present generality or state concretely which results in [9] are being quoted and why they extend verbatim to the non-integrable, gradient-term setting.","section":"Section 3, after Lemma 3.1"},{"comment":"The closedness of the continuity method is dispatched by invoking [10, Proposition 3.11] for the family (3.3). The hypotheses of that proposition are not checked. In particular, the right-hand side e^{ψ_t+c_t} in (3.3) depends on t, the constant c_t varies with t, and the operator contains the gradient terms Z(∂u). One also needs to know that the C_{e^{ψ_t+c_t},r,R}-subsolution produced in Proposition 3.8 is exactly the kind of subsolution required by [10, Proposition 3.11]. Since this estimate is the only argument preventing a breakdown of T at t = 1, the proof of Theorem 2.5, and hence of Theorem 1.3, remains incomplete until the applicability of [10, Proposition 3.11] is verified.","section":"Section 3, Lemma 3.11"}],"minor_comments":[{"comment":"The notation around (3.1)–(3.3) is confusing: ψ is used both for the given right-hand side of the equation and for log F of the chosen super-solution in the sentence 'Let ψ = log F[u]'. Using distinct symbols for the fixed data and the constructed functions would make the family (3.3) readable.","section":"Section 3, beginning of the proof"},{"comment":"Proposition 3.9(1) states F[u] = σe^{-ψ}. Since the sup-slope is defined as σ = inf_{u∈E} max_M e^{-ψ}F[u], the final solution satisfies e^{-ψ}F[u] = σ, so the formula should presumably be F[u] = σe^{ψ}; this looks like a sign typo.","section":"Section 3, Proposition 3.9"},{"comment":"There are numerous typographical errors, including 'funtion' in the abstract, 'unqiue' in Theorem 1.1, 'The the' in Theorem 2.5, 'coefficience' in the proof of Theorem 1.3, and 'Bejing' in the affiliation. These do not affect the mathematics but should be corrected in a revision.","section":"Throughout"},{"comment":"The sentence 'Also let σ be the sup-slope of the equation (1.6)' refers to equation (1.6), which is a specific application equation from Section 1, not the general equation (1.1) under consideration here. The reference should be to (1.1) (or to the abstract setting of Theorem 2.5).","section":"Section 3, after (3.1)"}],"recommendation":"major_revision","confidential_remarks":"This is a mathematical theorem paper with no data or code. The main issue is that the proof is conditional on unverified extensions of imported results; I would ask the editor to require the author to either prove the quoted lemmas in the present generality or spell out the exact hypotheses in [9] and [10] and verify them for the family (3.3). If that verification can be supplied, the paper is likely publishable, as the final twisted-constant formulas are explicit and address a natural open question. As it stands, the central claim is not fully established in the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper gives a genuine extension of the Guo–Song sup-slope criterion to equations with gradient terms on compact almost Hermitian manifolds, and the twisted-constant formulas in Theorem 1.3 are new and cleanly derived. The proof of the underlying criterion, though, delegates several key lemmas to [9] and one a priori estimate to [10]; those imports may well be fine, but the paper doesn't show it, and the referee should insist.\n\nWhat's actually new: the criterion itself for F[u]=e^ψ with ω_u = ω + √−1∂∂̄u + Z(∂u), and the resulting infimum formulas for the twisted constant in the form-type and Hessian-type Calabi-Yau equations. That answers a question from Chu–Tosatti–Weinkove. The reduction in the proof of Theorem 1.3 is a correct piece of linear algebra (the T transform), and the Hessian case is immediate from f=(S_k)^{1/k}. No circularity: the formulas are consequences of the criterion, not inputs.\n\nWhat the paper does well: it is short, the structure is clear, and the citation practice is honest—it builds directly on [9] and [10]. The extension is real, not a repackaging.\n\nSoft spots: the omitted proofs. After Lemma 3.1, it says 'Proofs of following lemmas can be found in [9], hence we omit them.' But Lemmas 3.2, Propositions 3.6 and 3.8 are used in a strictly more general setting: gradient terms, non-integrable complex structure, and a t-dependent right-hand side. The gradient terms enter only through the eigenvalues, so the arguments may carry over unchanged, but the author should say that or sketch the changes. Lemma 3.11 invokes [10, Prop. 3.11] for the a priori estimate; since [10] is precisely about gradient terms on almost Hermitian manifolds, I expect it applies, but the author must verify the hypotheses (uniform bounds on ψ_t and c_t, and the C-subsolution condition for the family (3.3)) rather than just citing. The statement of Theorem 2.5 is also sloppy: typos, missing spaces, and condition (4) is redundant. Fixable, but needs doing.\n\nI don't think the paper collapses; the stress-test's worry is a demand for verification rather than evidence of a gap. Still, as written, the proof is incomplete.\n\nWho it's for: complex differential geometers and PDE analysts working on Calabi-Yau type equations. It deserves a serious referee. I'd send it to review with the request that the author either prove the adapted lemmas or explicitly argue that the earlier proofs apply verbatim, and clean up Theorem 2.5. If that's done, the twisted-constant formulas are worth having.","headline":"Genuine new twisted-constant formulas for gradient-type equations on almost Hermitian manifolds, built on a sound criterion whose proof currently outsources the hard parts to earlier papers.","tokens_in":8708,"tokens_out":8636,"would_cite":true,"duration_ms":86930,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J05","32Q60","35J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Solvability of fully nonlinear elliptic equations on almost Hermitian manifolds is characterized by a sup-slope sub-solution condition, yielding exact infimum formulas for twisted constants.","keywords":["sup-slopes","sub-solutions","fully nonlinear elliptic equations","almost Hermitian manifolds","twisted constant","Calabi-Yau type equations","Hessian-type equations","form-type equations"],"falsifier":"If one can exhibit data satisfying the paper's hypotheses for which the deformed family $F[u+\\phi_t] = e^{\\psi_t+c_t}$ has solutions on $[0,1)$ with uniformly bounded $C^0$ norms but unbounded $C^2$ norms as $t \\to 1$, the closedness step and the equivalence fail. A more direct check: on a concrete non-integrable almost Hermitian manifold such as the six-sphere with a nearly Kähler structure, numerically solve the form-type equation and compare the resulting constant $c$ with the infimum formula in the main theorem.","tokens_in":7750,"feed_emoji":"🧮","tokens_out":8949,"duration_ms":85130,"temperature":0.7,"pith_summary":"The paper establishes that a broad class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, possibly containing gradient terms, is solvable exactly when a sub-solution in the admissible cone passes a pointwise comparison involving the sup-slope. The concrete payoff is a formula for the twisted constant, the additive constant forced by prescribing a volume form: it is the infimum over admissible functions of the maximum of the normalized operator. This answers a question raised in earlier work on the Monge-Ampère equation on non-integrable almost complex structures and unifies the classical, Hessian-type, and form-type Calabi-Yau equations. A reader should care because the formula converts an implicitly defined constant into a quantity that can in principle be computed or bounded from the data alone.","feed_headline":"Twisted constants computed by explicit infimum formulas","feed_subtitle":"For Calabi-Yau type equations, the forced additive constant is now an infimum formula instead of just a bound.","key_machinery":"The central object is the sup-slope $\\sigma = \\inf_{u\\in E} \\max_M e^{-\\psi}F[u]$, together with the limiting slope function $f_\\infty(\\lambda) = \\min_i \\lim_{R\\to\\infty} f(\\lambda_1,\\dots,R,\\dots,\\lambda_n)$ and the corresponding global sub-solution operator $F_\\infty[u] = f_\\infty(\\lambda(u))$. The criterion uses concavity of $f$ and the comparison $\\max_M e^{-\\psi}F[u] < \\min_M e^{-\\psi}F_\\infty[u]$ to run a continuity method through deformed equations $F[u+\\phi_t] = e^{\\psi_t+c_t}$; the $C^{r,R}$-subsolution bound $(\\lambda(\\omega_u) - r\\mathbf{1} + \\Gamma_n) \\cap \\partial\\Gamma_h \\subset B(0,R)$ turns these inequalities into uniform a priori estimates, closing the method.","core_discovery":"On a compact almost Hermitian manifold, write $\\omega_u = \\omega + \\sqrt{-1}\\partial\\bar\\partial u + Z(\\partial u)$. The paper proves that the equation $F[u] = e^{\\psi}$ with admissible cone $\\Gamma$ has a smooth solution if and only if there exists $u \\in E$ with $e^{-\\psi}F_\\infty[u] > \\sigma$, where $\\sigma = \\inf_{u\\in E} \\max_M e^{-\\psi}F[u]$ and $F_\\infty$ is built from the limiting slopes of $f$. When a solution exists it is unique up to additive constant and satisfies $F[u] = \\sigma e^{\\psi}$. For the form-type equation with gradient terms, this gives $e^c = \\inf_u \\max_M e^{-\\psi}\\left(\\eta + \\tfrac{1}{n-1}((\\Delta_C u)\\chi - \\sqrt{-1}\\partial\\bar\\partial u) + W(\\partial u)\\right)^n/\\chi^n$; for the $k$-Hessian type equation, $e^c = \\inf_u \\max_M e^{-\\psi}\\,\\omega_u^k \\wedge \\chi^{n-k}/\\chi^n$.","pith_inferences":["Beyond the paper: the infimum formulas suggest a numerical route to twisted constants: optimize $e^{-\\psi}F[u]$ over a finite-dimensional family of admissible functions, and the paper's equivalence implies the optimal value converges to $e^c$ whenever a solution exists.","Beyond the paper: if the quoted a priori estimates are valid for the deformed family used here, the same sup-slope criterion would likely extend to other structure functions $f$ whose associated limiting function $f_\\infty$ remains concave and whose $C$-subsolution level sets stay bounded.","Beyond the paper: the linear eigenvalue transformation used for the form-type equation shows that the twisted constant problem with gradient terms is equivalent, through that transform, to a no-gradient Hessian-type problem, which may allow transferring regularity results between the two settings.","Beyond the paper: a direct check on a non-integrable example with known solutions could test whether the infimum in the main theorem is attained by a smooth solution, a question the paper does not address."],"forward_implications":["Solvability of the equation is reduced to a checkable inequality, namely the existence of a sub-solution satisfying $e^{-\\psi}F_\\infty[u] > \\sigma$, rather than requiring an explicit solution construction.","The twisted constant $c$ in both the form-type equation and the Hessian-type equation is exactly the infimum formula in the main theorem, replacing the previous bound $|c| \\le \\sup_M |F|$ with an exact value.","Any smooth solution is unique up to an additive constant and the normalized operator $F[u]/e^{\\psi}$ is exactly the constant $\\sigma$, so the sup-slope encodes the normalization of the solution.","The sup-slope scales explicitly under shifting $\\psi$: $\\sigma(\\psi + C) = e^{-C}\\sigma(\\psi)$, which also fixes the corresponding shift of $c$.","The method gives a unified treatment of the classical Monge-Ampère equation, the Hessian-type equations, and the form-type equations with gradient terms on compact almost Hermitian manifolds."],"supporting_citations":[{"why":"Supplies the sup-slope and sub-solution method, including the comparison lemmas that this paper adapts to gradient terms and to almost Hermitian manifolds.","marker":"[9]"},{"why":"Provides the a priori estimates on compact almost Hermitian manifolds with gradient terms, used to close the continuity method.","marker":"[10]"},{"why":"Proved the Monge-Ampère equation for non-integrable almost complex structures and raised the question of determining the twisted constant.","marker":"[3]"},{"why":"Introduced the C-subsolution notion and the boundedness criterion used to convert the sub-solution comparison into uniform estimates.","marker":"[13]"},{"why":"Extended solvability of fully nonlinear elliptic equations to compact almost Hermitian manifolds, providing the framework for equations with gradient terms.","marker":"[2]"},{"why":"Solved the form-type equation with gradient terms on Hermitian manifolds, the result whose twisted constant is now determined in the almost Hermitian case.","marker":"[14]"}],"fun_headline_variants":["Twisted constants are explicit infima for Calabi-Yau type equations","Calabi-Yau type equations: twisted constants via explicit infima","Infimum formula gives explicit twisted constants for Calabi-Yau type equations","Twisted constants become explicit infima in Calabi-Yau type equations","Explicit infimum formulas pin down twisted constants in Calabi-Yau type equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the a priori estimates and sub-solution comparison lemmas from a simpler setting, without gradient terms and with an integrable complex structure, carry over unchanged to equations with gradient terms on non-integrable almost Hermitian manifolds; the paper quotes these results instead of proving them.","fun_headline_variants_meta":{"raw":{"variants":["Twisted constants are explicit infima for Calabi-Yau type equations","Calabi-Yau type equations: twisted constants via explicit infima","Infimum formula gives explicit twisted constants for Calabi-Yau type equations","Twisted constants become explicit infima in Calabi-Yau type equations","Explicit infimum formulas pin down twisted constants in Calabi-Yau type equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2742,"prompt_tokens":874,"completion_tokens":1868,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":490,"completion_tokens_details":{"reasoning_tokens":1771}},"tokens_in":490,"tokens_out":1868,"duration_ms":11737,"temperature":1.0,"reasoning_tokens":1771,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:43:25.669661+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If one can exhibit data satisfying the paper's hypotheses for which the deformed family $F[u+\\phi_t] = e^{\\psi_t+c_t}$ has solutions on $[0,1)$ with uniformly bounded $C^0$ norms but unbounded $C^2$ norms as $t \\to 1$, the closedness step and the equivalence fail. A more direct check: on a concrete non-integrable almost Hermitian manifold such as the six-sphere with a nearly Kähler structure, numerically solve the form-type equation and compare the resulting constant $c$ with the infimum formula in the main theorem.","supporting_citations":[{"cited_title":"Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds , Math","cited_arxiv_id":null,"evidence_quote":"Provides the a priori estimates on compact almost Hermitian manifolds with gradient terms, used to close the continuity method."},{"cited_title":"The Monge-Amp` ere equation for non-integrable almost complex structures, J","cited_arxiv_id":null,"evidence_quote":"Proved the Monge-Ampère equation for non-integrable almost complex structures and raised the question of determining the twisted constant."},{"cited_title":"Differential Geom","cited_arxiv_id":null,"evidence_quote":"Introduced the C-subsolution notion and the boundedness criterion used to convert the sub-solution comparison into uniform estimates."},{"cited_title":"Fully nonlinear elliptic equations on compact almost Hermitian mani- folds, Calc","cited_arxiv_id":null,"evidence_quote":"Extended solvability of fully nonlinear elliptic equations to compact almost Hermitian manifolds, providing the framework for equations with gradient terms."},{"cited_title":"Gauduchon metrics with prescribed volume form , Acta Math","cited_arxiv_id":null,"evidence_quote":"Solved the form-type equation with gradient terms on Hermitian manifolds, the result whose twisted constant is now determined in the almost Hermitian case."}],"review_version":1}