{"id":"33dec266-521c-4768-9f75-5c5686271304","arxiv_id":"2506.16305","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of solutions for fully nonlinear elliptic equations on compact almost Hermitian manifolds is established under a sub-slope condition, with applications to the Hessian quotient and deformed Hermitian-Yang-Mills equations.","lead":"This math paper proves that a broad class of fully nonlinear elliptic equations can be solved on compact almost Hermitian manifolds whenever a certain technical 'subsolution' condition holds. It applies this to two geometric equation families, the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation, in a setting where existence was previously open.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The closedness step of Theorem 3.1 applies Theorem 2.4 to φ_t, but Theorem 2.4 as quoted requires the solution itself to be a C-subsolution; if [12] does not give estimates for arbitrary solutions with a separate C-subsolution, the existence proof collapses.","rationale":"I read the paper in good faith. The comparison principle, the sub-slope inequalities in Lemma 3.2 and Proposition 3.4, and the Fredholm/openness argument are standard and appear repairable; the main existence claim therefore hinges on the a priori estimate. The paper's Theorem 2.4 is quoted with the same symbol u for the C-subsolution and the solution; the proof of closedness in Section 3.1 needs the estimate for the solution φ_t while only the fixed function u is known to be a C-subsolution. This is exactly the kind of external input that cannot be verified from the manuscript alone. If the cited theorem has the usual separate-subsolution form, the paper's conclusion follows; if not, there is a genuine gap. This does not change the reader's conditional verdict, but it makes the check of [12] the decisive step. Other concerns (notation in Theorems 1.1–1.3, the dHYM constant σ in (1.8), the normalization in the openness argument) are real but secondary.","tokens_in":8186,"tokens_out":15337,"duration_ms":162518,"concrete_test":"Retrieve Huang-Zhang [12, Corollary 1.4 and Proposition 3.11] and check the exact hypotheses: (a) is the estimate stated as “for every solution v with a C-subsolution u” or as “for a function that is both a subsolution and a solution”? (b) Does the estimate apply to equations of the family (3.5), i.e., F(ω_v) = h_t + c_t with t-dependent RHS and with Z(∂v) gradient terms, and to the dHYM phase equation (1.6)? If (a) is the former and (b) holds, the closedness step is sound after correcting the quotation; otherwise Theorem 3.1 lacks a proof and the applications in Theorems 1.2–1.3 are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 proves that the fixed function u is a C-subsolution of the deformed family (3.5), then uses “Proposition 3.5 and Theorem 2.4” to conclude the a priori bound ‖φ_t‖_{C^{2,α}} ≤ C and hence closedness. But Theorem 2.4, as stated on p. 6, reads “Suppose that u is a C-subsolution … Suppose that u is a smooth solution … Then ‖u‖…” — the same function is required to be both the subsolution and the solution. In the continuity argument the solution is φ_t (or ¯u + φ_t), not u; nothing in the paper shows that the solution is itself a C-subsolution. The standard formulation of Székelyhidi-type estimates allows a separate C-subsolution v and yields control of any solution u, but the present quotation does not say that. If [12, Cor. 1.4/Prop. 3.11] is indeed of the separate-subsolution form, the gap is only a misstatement; if it requires the solution to be a C-subsolution, then the closedness argument is invalid. The dHYM case is only asserted to be “similar,” so the same verification must cover (1.6).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the sub-slope method of Guo-Song to compact almost Hermitian manifolds and claims an existence theorem for fully nonlinear elliptic equations of the form F(ω_u)=h+σ, including equations with gradient terms. The main result, Theorem 1.1, states an equivalence between solvability, existence of a C-subsolution, and a sub-slope inequality. The proof uses a continuity method, with closedness relying on a priori estimates quoted from Huang-Zhang and openness obtained by a Fredholm alternative argument. Applications are stated for the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting.","tokens_in":8456,"tokens_out":12947,"duration_ms":152224,"significance":"If the main existence theorem is correct, the paper gives a practically checkable criterion for solvability of a broad class of fully nonlinear equations on almost Hermitian manifolds, and it would provide the first existence results for the complex Hessian quotient equation and the dHYM equation in that category. The paper is concise and builds on substantial prior work: the C-subsolution framework of Székelyhidi, the sub-slope idea of Guo-Song, and the a priori estimates of Huang-Zhang. The overall strategy is natural and the openness argument is standard. However, several load-bearing points are not adequately supported in the manuscript as written: the quoted a priori estimate is applied in a form that does not match the continuity argument, the claimed equivalence in Theorem 1.1 is only proved in one direction, and the dHYM application is dismissed with a 'similar' argument despite failing one of the structural hypotheses. These issues are local and likely fixable, so the paper merits revision rather than rejection.","major_comments":[{"comment":"The equivalence in Theorem 1.1 is not established. Section 3 proves (2) equivalent to (3) via Proposition 2.2 and then proves Theorem 3.1, which is the implication (3) to (1). The converse implication (1) to (3) is never proved. Lemma 3.7 only shows that a solution u of F(u)=h+σ satisfies σ=max_M(F(u)-h); this is an identity for a solution, not a proof that F∞(ω_u)>h+σ. If the intended argument is that every solution is automatically a C-subsolution, that fact should be stated and proved, for example by showing f∞>f on Γ under assumptions (i)-(iii) and for the dHYM phase. Otherwise Theorem 1.1 should be reformulated as a sufficiency theorem.","section":"Theorem 1.1 and Section 3"},{"comment":"The closedness of the continuity set applies Theorem 2.4 to φ_t, but the theorem as quoted on p.6 requires the same function to be both a C-subsolution and a solution. In the continuity path (3.5), the fixed function u is shown in Proposition 3.5 to be a C-subsolution, while the function whose C^{2,α} norm is needed is φ_t, or equivalently \\bar u+φ_t. If Huang-Zhang [12, Cor. 1.4/Prop. 3.11] is the standard separate-subsolution estimate, then Theorem 2.4 must be restated with two distinct functions and the hypotheses for the deformed family (3.5) must be verified, including the gradient terms and the dHYM case. If the estimate really requires the solution itself to be a C-subsolution, then the bound \\|φ_t\\|\\le C does not follow and the closedness step collapses.","section":"Section 3.1, application of Theorem 2.4"},{"comment":"The constants appearing in the two applications are not stated in the same normalization as Definition 2.2. In Theorem 1.2 the equation (1.4) contains σ in the exponent, while (1.5) defines log σ as the infimum of max(log(ω_{u'}^k∧χ^{n-k}/(ω_{u'}^l∧χ^{n-l}))-h); thus the σ in (1.4) is the exponential of the sub-slope of Definition 2.2, not the sub-slope itself. In Theorem 1.3, (1.8) defines tan σ, again a different normalization, and the minimum is taken over a restricted phase interval. If these formulas are intended as explicit computations of the sub-slope for the two equations, the equivalence should be stated and proved; otherwise the constant in the existence theorem is ambiguous.","section":"Theorems 1.2 and 1.3, equations (1.5) and (1.8)"},{"comment":"The dHYM case is treated by the sentence 'the proof is similarly to Theorem 1.1' (p.4), but Theorem 3.1 and its proof rely on condition (iii), on the sub-slope definition through inf max(F-h), and on the strict inequality in (3.4). The dHYM phase function does not satisfy condition (iii), and the sub-slope in (1.8) is defined through tan and a restricted phase interval rather than through Definition 2.2. The manuscript should either carry out the dHYM adaptation lemma by lemma, or explicitly identify which of the claims (3.3)-(3.8) remain valid and why. As written, Theorem 1.3 is not proved.","section":"Theorem 1.3 and Section 3"}],"minor_comments":[{"comment":"The title reads 'FULL Y NON-LINEAR' and should read 'FULLY NON-LINEAR'; there are also typos such as 'seting' on p.3 and 'SUpposethat' on p.8.","section":"Title and page 2"},{"comment":"In the first paragraph the citation list '[5, 6, 12, 11, 12]' contains duplicate [12] and is out of order; please correct.","section":"Introduction, references"},{"comment":"Lemma 3.7 refers to 'the sup-slope' although Definition 2.2 defines the sub-slope; either the terminology should be consistent or the intended distinction should be explained.","section":"Lemma 3.7"},{"comment":"The introduction states that Theorem 1.2 will be proved in Section 3, but Section 3 ends after the lemmas for Theorem 3.1; a short verification that the Hessian quotient equation satisfies conditions (i)-(iii) and that (1.5) is the corresponding sub-slope should be added.","section":"Section 3, proof of Theorem 1.2"},{"comment":"The letter u is overloaded: in Theorem 2.4 it denotes both the subsolution and the solution, and in Section 3.1 it denotes the fixed admissible function while the solution is \\bar u+φ_t. Using distinct symbols such as \\underline u and v would prevent the ambiguity that is currently at the center of the closedness argument.","section":"Theorem 2.4 and notation"}],"recommendation":"major_revision","confidential_remarks":"This is a short paper whose main content is an adaptation of Guo-Song's sub-slope method to the almost Hermitian setting, relying heavily on the author's previous estimates with Zhang. The overlap with the cited preprint by Lin is acknowledged but not discussed in detail; I would ask the editor to ensure the revised version clarifies the relation to [17]. The technical gaps are, in my assessment, likely repairable: the misquotation of the separate-subsolution estimate is probably fixable, and the dHYM case requires a genuine but feasible verification. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The genuinely new contribution is the transfer of Guo-Song's sub-slope technique to compact almost Hermitian manifolds, including gradient terms, and the first existence statements for the complex Hessian quotient equation and dHYM in that setting. If the machinery works, that's a real gap filled.\n\nThe main sufficiency theorem (Theorem 3.1) follows the familiar continuity method: define sub-slope as an infimum, prove the subsolution is preserved along the path, then get closedness from the cited a priori estimates. The argument is concise and mostly standard.\n\nThe soft spots are real but mostly presentational. First, Theorem 1.1 states an equivalence but only proves (3)⇒(1). The reverse direction is not automatic—a solution of F(u)=h+σ need not satisfy the strict inequality σ < min(f∞(u)-h) that defines a C-subsolution. This overclaim should be corrected; the applications only need sufficiency.\n\nSecond, the definitions of σ in Theorems 1.2 and 1.3 (displayed in (1.5) and (1.8)) are not well-posed as written. The variable u' ranges over an unspecified set, and the expressions are missing the normalization by χ^n. These are fixable, but as written the statements are ill-defined.\n\nThird, and more substantively, the closedness step in Section 3.1 invokes Theorem 2.4 on the solution φ_t while the C-subsolution is the fixed u. The theorem as quoted says 'u is a C-subsolution ... Suppose u is a smooth solution ...', which suggests the same function must play both roles. In the continuity argument they are different. If Huang-Zhang's estimate is of the standard separate-subsolution form, this is a misstatement; if not, the argument has a genuine gap. The paper should either quote the theorem correctly or prove that the solution is itself a C-subsolution along the path.\n\nThe dHYM application is only sketched; since dHYM does not satisfy condition (iii), the 'similar' proof deserves a few checks.\n\nOverall, the core idea is sound and the paper deserves a serious referee. I would send it to review, but with a request for careful revision. In its current form I would not cite it, because the statements need fixing. After revision, it could be a useful paper for people working on fully nonlinear PDEs on almost complex manifolds.","headline":"Plausible extension of Guo-Song sub-slope to almost Hermitian manifolds with new existence results for Hessian quotient and dHYM, but the presentation has several statement-level issues that need fixing.","tokens_in":8969,"tokens_out":4270,"would_cite":false,"duration_ms":47822,"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":"This paper proves that a sub-slope inequality is necessary and sufficient for existence of smooth solutions to a broad class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, and derives applications to two…","keywords":["Fully non-linear elliptic equations","sub-slope","C-subsolution","Almost Hermitian manifold","Complex Hessian quotient equation","Deformed Hermitian-Yang-Mills equation","Continuity method","A priori estimates"],"falsifier":"A concrete check would be to take a compact almost Hermitian 4-manifold with data satisfying conditions (i)-(iii) and a genuine C-subsolution, then run the continuity path numerically: if second derivatives of $\\varphi_t$ blow up while the sub-slope inequality holds, Theorem 3.1 would be false. Alternatively, a counterexample to the a priori estimate of [12] inside the dHYM or gradient-term class would collapse the closedness step directly.","tokens_in":7971,"feed_emoji":"📐","tokens_out":10718,"duration_ms":105821,"temperature":0.7,"pith_summary":"This paper establishes a general existence principle for fully nonlinear elliptic equations of the form $F(\\omega_u)=h+\\sigma$ on compact almost Hermitian manifolds, where $\\omega_u$ is a deformed $(1,1)$-form depending on the unknown function and its gradient. The principle is that solvability is equivalent to a sub-slope condition: there exists an admissible function $u$ with $\\sigma < \\min(F_\\infty(u)-h)$, where $\\sigma$ is the infimum over admissible functions of the largest gap $F(u)-h$. The author generalizes the sub-slope method from Hermitian to almost Hermitian manifolds and drops the earlier positivity assumption on the right-hand side. As applications, the paper proves existence of smooth solutions for the complex Hessian quotient equation and for the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting. A reader would care because these equations are central in complex geometry, and the almost Hermitian case had resisted the previous existence arguments.","feed_headline":"Sub-slope condition yields solutions on almost Hermitian manifolds","feed_subtitle":"A single sub-slope inequality controls existence of smooth solutions for two geometric PDEs.","key_machinery":"The sub-slope is $\\sigma=\\inf_{u\\in E}\\max_M(F(u)-h)$, with $E$ the set of functions whose eigenvalue tuple lies in the cone $\\Gamma$. The asymptotic function $f_\\infty(\\lambda)=\\min_i \\lim_{R\\to\\infty} f(\\lambda_1,\\dots,R,\\dots,\\lambda_n)$ characterizes C-subsolutions: $\\bar u$ is a C-subsolution exactly when $f_\\infty(\\lambda(\\omega_{\\bar u}))>h$. The continuity method deforms the right-hand side by $h_t=(1-t)F(\\bar u)+th$ and tracks a constant $c_t$; the sub-slope inequality ensures the subsolution property survives, the a priori estimate closes the set of good parameters, and the Fredholm alternative opens it. This machinery replaces the previously required positivity of the right-hand side and works for the deformed Hermitian-Yang-Mills phase equation as well.","core_discovery":"The central claim is Theorem 3.1: if there exists $u\\in E$ with $\\sigma<\\min(F_\\infty(u)-h)$, then the equation $F(\\omega_u)=h+\\sigma$ has a smooth solution $u\\in E$. The proof runs a continuity path $F(\\bar u+\\varphi_t)=h_t+c_t$ between an already-solved equation for $\\bar u$ and the target equation; the sub-slope inequality keeps $\\bar u$ a C-subsolution at every step, the a priori estimates give uniform $C^{2,\\alpha}$ control, and the Fredholm alternative gives openness. A later maximum-principle argument forces the final constant $c_t$ to equal the sub-slope $\\sigma$. The paper also treats the deformed Hermitian-Yang-Mills equation, whose $f$ does not satisfy the growth condition (iii), by noting that the same subsolution-preservation argument goes through. The geometric consequences are Theorem 1.2 for the complex Hessian quotient equation and Theorem 1.3 for the deformed Hermitian-Yang-Mills equation.","pith_inferences":["The equivalence statement turns the existence problem into a check of one scalar inequality, so explicit almost Hermitian examples such as solvmanifolds could be tested for the sub-slope threshold against known obstructions.","Because the proof only needs the subsolution to survive along the path, the same scheme should transfer to other equations whose $f_\\infty$ is bounded, including parabolic versions of these equations, once a $C^{2,\\alpha}$ estimate is available.","Dropping the positivity of the right-hand side is likely to matter for geometric applications with indefinite background forms or sign-changing $h$, where earlier subsolution arguments did not apply.","The paper's treatment of gradient terms through $Z(\\partial u)$ suggests the result may cover more general Lagrangian-type equations on almost complex manifolds if they can be cast in this form."],"forward_implications":["For data satisfying conditions (i)-(iii), the three statements in Theorem 1.1 become equivalent: a smooth solution exists, a C-subsolution exists, and the sub-slope inequality $\\sigma<\\min(F_\\infty(u)-h)$ holds.","The complex Hessian quotient equation (1.4) on compact almost Hermitian manifolds admits a unique smooth solution whenever a C-subsolution exists.","The deformed Hermitian-Yang-Mills equation (1.6) with $h$ in the supercritical range admits a unique smooth solution on compact almost Hermitian manifolds whenever a C-subsolution exists.","The constant produced by the continuity method is forced by the maximum principle to be exactly the sub-slope $\\sigma$, so the method solves the original equation with the threshold right-hand side, not merely some nearby constant."],"supporting_citations":[{"why":"Introduces the sub-slope and the continuity method that this paper adapts to almost Hermitian manifolds.","marker":"[8]"},{"why":"Supplies the a priori $C^{2,\\alpha}$ estimates for equations with gradient terms on almost Hermitian manifolds, the load-bearing estimate for closedness.","marker":"[12]"},{"why":"Introduces the C-subsolution condition and its characterization through $f_\\infty$.","marker":"[25]"},{"why":"Proves the earlier $C^{2,\\alpha}$ estimates for nonlinear elliptic equations on almost Hermitian manifolds without gradient terms.","marker":"[5]"},{"why":"Gives the dHYM existence result under C-subsolutions on Kähler manifolds that the almost Hermitian version extends.","marker":"[3]"},{"why":"Provides a priori estimates for Hessian quotient equations in the almost Hermitian setting.","marker":"[29]"},{"why":"Solves the complex Hessian quotient equation on Kähler manifolds for general $k$ and $l$, one of the cases extended here.","marker":"[7]"},{"why":"Treats the J-equation on Kähler manifolds, the $k=n$, $l=n-1$ special case of the Hessian quotient equation.","marker":"[20]"}],"fun_headline_variants":["Sub-slope inequality yields smooth solutions on almost Hermitian manifolds","Sub-slope solves Hessian quotient and deformed Hermitian-Yang-Mills","Existence via sub-slope on almost Hermitian manifolds","Sub-slope condition solves two geometric PDEs on almost Hermitian manifolds","Almost Hermitian sub-slope ensures PDE solvability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole existence result leans on the a priori $C^{2,\\alpha}$ estimate for equations like (1.1) and (1.6) on compact almost Hermitian manifolds; if that estimate fails for some allowed data, the continuity-path closedness argument breaks.","fun_headline_variants_meta":{"raw":{"variants":["Sub-slope inequality yields smooth solutions on almost Hermitian manifolds","Sub-slope solves Hessian quotient and deformed Hermitian-Yang-Mills","Existence via sub-slope on almost Hermitian manifolds","Sub-slope condition solves two geometric PDEs on almost Hermitian manifolds","Almost Hermitian sub-slope ensures PDE solvability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001197,"raw_usage":{"total_tokens":4870,"prompt_tokens":812,"completion_tokens":4058,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":3966}},"tokens_in":428,"tokens_out":4058,"duration_ms":31863,"temperature":1.0,"reasoning_tokens":3966,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:44:53.829847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to take a compact almost Hermitian 4-manifold with data satisfying conditions (i)-(iii) and a genuine C-subsolution, then run the continuity path numerically: if second derivatives of $\\varphi_t$ blow up while the sub-slope inequality holds, Theorem 3.1 would be false. Alternatively, a counterexample to the a priori estimate of [12] inside the dHYM or gradient-term class would collapse the closedness step directly.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the a priori $C^{2,\\alpha}$ estimates for equations with gradient terms on almost Hermitian manifolds, the load-bearing estimate for closedness."},{"cited_title":"Fully non-linear elliptic equations on compact Hermitian m anifolds, J","cited_arxiv_id":null,"evidence_quote":"Introduces the C-subsolution condition and its characterization through $f_\\infty$."},{"cited_title":"Zhang, J","cited_arxiv_id":null,"evidence_quote":"Proves the earlier $C^{2,\\alpha}$ estimates for nonlinear elliptic equations on almost Hermitian manifolds without gradient terms."},{"cited_title":"(1,1) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions, Camb","cited_arxiv_id":null,"evidence_quote":"Gives the dHYM existence result under C-subsolutions on Kähler manifolds that the almost Hermitian version extends."},{"cited_title":"Monge-Amp\\`{e}re type equations on almost Hermitian manifolds","cited_arxiv_id":"2101.00380","evidence_quote":"Provides a priori estimates for Hessian quotient equations in the almost Hermitian setting."},{"cited_title":"On a class of fully nonlinear flows in K¨ ahler geometry , J","cited_arxiv_id":null,"evidence_quote":"Solves the complex Hessian quotient equation on Kähler manifolds for general $k$ and $l$, one of the cases extended here."},{"cited_title":"On the convergence and singularities of the J-flow with appli- cations to the Mabuchi energy , Comm","cited_arxiv_id":null,"evidence_quote":"Treats the J-equation on Kähler manifolds, the $k=n$, $l=n-1$ special case of the Hessian quotient equation."}],"review_version":1}