{"id":"c2b5d3e2-7845-4a9c-848d-91f35ee515f1","arxiv_id":"2607.19132","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite-weighted-energy m-subharmonic functions on a quasi-m-hyperconvex domain in a compact Kähler manifold extend maximally to the whole manifold with controlled complex Hessian measure and weighted energy.","lead":"This paper proves an extension theorem for a family of generalized subharmonic functions on curved complex spaces: under mild volume conditions, any finite-weighted-energy function defined on a subdomain can be expanded to the whole space while keeping its energy and Hessian mass under control. The result generalizes a known plurisubharmonic theorem to the m-subharmonic setting and is read mainly by specialists in complex analysis and Kähler geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Energy-preservation step in Thm 3.4 rests on an unstated convergence theorem from [CN] whose hypotheses and conclusion are not verified.","rationale":"The reader's CONDITIONAL verdict identifies the same cluster of concerns: the proof leans on black-box results from [CN] without precise statements. I agree that the paper's argument is probably fixable, but the single most load-bearing point is the convergence theorem used in Theorem 3.4(i). If that theorem does not supply the needed weighted-measure limit inequality, then the main theorem's assertion φ̃∈E_χ^m(X,ω) and the energy inequality are not established from the given proof. The existence theorem [CN, Theorem 1.3] is also load-bearing, but it is more plausibly applicable to the patched measures; the convergence theorem is invoked in a context (weighted Hessian measures on a compact Kähler manifold, decreasing limit of subextensions) that the paper does not define or prove. Since this is a fixable gap rather than a demonstrated falsehood, the verdict should remain CONDITIONAL, not ACCEPT and not REJECT. My recommendation is UNCHANGED relative to the reader's verdict.","tokens_in":21186,"tokens_out":27211,"duration_ms":233098,"concrete_test":"Locate the exact statements of the two results from [CN] used: Theorem 1.3 (existence for H_m(u)=μ) and the convergence theorem for weighted energy classes on compact Kähler manifolds. Check: (a) whether Theorem 1.3 applies verbatim to μ_j = 1_Ω H_m(φ_j)+ε_j ω^n, including the normalization sup_X u_j = −1; (b) whether the convergence theorem's hypotheses are met by the sequence φ̃_j, in particular the uniform bound ∫_X −χ(φ̃_j)H_m(φ̃_j) ≤ C; (c) whether its conclusion gives the measure inequality −χ(φ̃)1_Ω H_m(φ̃) ≤ −χ(φ)1_Ω H_m(φ), not merely convergence of total weighted energies. If either result is not stated in [CN] or requires extra conditions, Theorem 3.4 is incomplete as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.4(i) — the main theorem's energy-preservation claim — is carried by an unstated ``convergence theorem in [CN]'' invoked after φ̃_j ↘ φ̃ on X. To conclude φ̃ ∈ E_χ^m(X,ω) and the measure inequality −χ(φ̃)1_Ω H_m(φ̃) ≤ −χ(φ)1_Ω H_m(φ), one needs a theorem asserting that weighted Hessian measures pass to decreasing limits in this setting. This is stronger than the usual weak compactness of Hessian measures or semicontinuity of total weighted energy: it requires that the measures −χ(φ̃_j) H_m(φ̃_j) converge to −χ(φ̃) H_m(φ̃), that −χ(φ_j)1_Ω H_m(φ_j) converge to −χ(φ)1_Ω H_m(φ), and that the order inequality survives the limit. No such statement is included, and the paper's own Theorem 2.25 is proved only for domains and for approximating sequences in E_0^m(Ω,ω), not for the φ̃_j on X. If [CN]'s convergence theorem has additional hypotheses — e.g., a growth condition on χ, a uniform capacity bound, or a requirement that the approximants already lie in E_χ^m(X,ω) — those hypotheses are not checked. The same gap affects the support argument in (iii), which also relies on passing a weighted measure inequality to the limit and uses undefined truncations φ_s, φ_t. This is a load-bearing presentation and verification gap, not merely a stylistic omission.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that if Ω is a quasi-m-hyperconvex domain in a compact Kähler manifold (X,ω) with ∫_Ω ω^n < ∫_X ω^n, and χ is a convex weight function, then every φ in the weighted Hessian energy class E_χ^m(Ω,ω) with ∫_Ω H_m(φ) ≤ ∫_X ω^n admits a maximal ω-m-subharmonic subextension φ̃ to X. The main theorem asserts that φ̃ ∈ E_χ^m(X,ω), the weighted energy does not increase, the local Hessian domination 1_Ω H_m(φ̃) ≤ 1_Ω H_m(φ) holds, and H_m(φ̃) is supported on {φ̃ = φ} ∪ ∂Ω. The proof follows the CKZ2 strategy: approximate φ by functions in E_0^m, construct approximate subextensions via the solvability of degenerate complex Hessian equations on X, and pass to decreasing limits. Section 4 adapts the result to C^n via the Fubini–Study correspondence and introduces an m-Lelong class.","tokens_in":21510,"tokens_out":6065,"duration_ms":58222,"significance":"If the proof is completed, this would be a natural and useful extension of the CKZ2 maximal-subextension theorem to m-subharmonic functions and weighted energy classes, with potential applications to complex Hessian operators on compact Kähler manifolds. The paper contains coherent comparison principles, mass estimates, and approximation arguments, and the statement of the main theorem is precise and falsifiable. The main weakness is that the decisive limit passage in Theorem 3.4 is delegated to an unnamed convergence theorem, and the support argument uses undefined approximations; these are load-bearing verification gaps rather than stylistic issues.","major_comments":[{"comment":"The proof states: \"Since φ̃_j ↘ φ̃ on X, the convergence theorem in [CN] implies φ̃ ∈ E_χ^m(X,ω)\" and gives the weighted measure inequality. No such theorem is stated, and the paper's own Theorem 2.25 is proved only for domains Ω and for approximating sequences in E_0^m(Ω,ω); the φ̃_j are bounded ω-m-subharmonic functions on X without vanishing boundary data. To conclude the energy-preservation claim, one needs a precise convergence theorem for weighted Hessian measures along decreasing sequences on X, including convergence of −χ(φ̃_j)H_m(φ̃_j) to −χ(φ̃)H_m(φ̃), convergence of −χ(φ_j)1_ΩH_m(φ_j), and preservation of the measure inequality. Please state the theorem from [CN], verify its hypotheses for the present sequence, or supply a proof. This is the central claim (i) of the main theorem.","section":"Theorem 3.4(i)"},{"comment":"The support statement uses several undefined objects. The proof reuses the symbol φ_j for the truncations max{φ,−j}, conflicting with the approximating sequence φ_j ∈ E_0^m(Ω,ω) used earlier. Later \"for fixed s and t\" the functions φ_s and φ̃_s are not defined. The passage 1_{φ>−j}H_m(φ_j) ↗ 1_{φ>−∞}H_m(φ) is asserted by reference to [CN] without stating hypotheses, and Lemma 2.11 is invoked without checking the required uniform capacity control. Since claim (iii) is part of the main theorem, these gaps must be fixed.","section":"Theorem 3.4(iii)"},{"comment":"The construction of the approximate subextensions rests on the black-box solvability statement [CN, Theorem 1.3], used to produce u_j with H_m(u_j) = 1_Ω H_m(φ_j) + ε_j ω^n and sup_X u_j = −1. The exact hypotheses of that theorem are not stated. In particular, the paper should verify that 1_Ω H_m(φ_j) + ε_j ω^n does not charge m-polar sets and satisfies whatever integrability or regularity conditions [CN, Theorem 1.3] requires. If the theorem needs additional hypotheses beyond \"does not charge m-polar sets\", the approximation scheme collapses and the existence of φ̃ is not established.","section":"Theorem 3.2"}],"minor_comments":[{"comment":"There are numerous typos and notation slips: Theorem 3.2 says \"quasi-hyperconvex\" instead of \"quasi-m-hyperconvex\"; Proposition 2.17(3) has ω^{m−n} where it should be ω^{n−m}; the abstract says \"a good control properties\"; Proposition 3.3 uses \"eφ\" and \"eφ_j\" for φ̃ and φ̃_j.","section":"Throughout"},{"comment":"The proof begins \"Since P(f)^* ≤ f, then P(f) = P(f)^*\", but P(f) is not known to be upper semicontinuous before the proof. This step needs justification or rewording.","section":"Theorem 2.9"},{"comment":"The symbol \\hat φ appears before the subextension φ̃ is introduced, and the argument with the characteristic function χ_K should specify that K is compact (hence closed) and explain the approximation by continuous functions more carefully.","section":"Lemma 3.1"},{"comment":"The hypothesis says H_m(u) \"charges no pluripolar sets\", while elsewhere the relevant notion is m-polar sets. Please align the terminology and clarify whether the stronger condition is needed in the C^n-to-P^n transfer.","section":"Theorem 4.3"},{"comment":"The decomposition H_m(φ̃) = f · 1_Ω H_m(φ) + ν with ν supported on ∂Ω is asserted without saying with respect to which reference measure the density f is taken. A short explanation would improve readability.","section":"Remark 3.5"}],"recommendation":"major_revision","confidential_remarks":"The paper has a clear and plausible plan, and the main theorem is likely true. However, the current version leaves a central convergence step as an unstated black box and uses undefined truncations in the support argument. I would encourage a revision that states and proves (or precisely cites and verifies) the needed convergence theorem for weighted Hessian measures on X, and that cleans up Theorem 3.4(iii). Once those gaps are closed, the paper would be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: this is a genuine extension of the Cegrell–Kołodziej–Zeriahi maximal subextension result to weighted energy classes for m-subharmonic functions on compact Kähler manifolds. The main theorem is not in the cited literature, and the paper does the expected work of adapting the CKZ2 architecture: maximal subextension defined by upper envelope, approximation by bounded functions, Hessian mass estimates, support on the contact set. The Hessian-measure comparison in Lemma 3.1 and the capacity estimate in Lemma 2.11 are standard but carefully written. Credit where due: the mass estimates in Section 3 are plausible, and the authors correctly identify the degenerate complex Hessian equation [CN] as the black box. This is a technical advance, not a breakthrough, but a real one.\n\nThe soft spots are concentrated in Theorem 3.4, and they are load-bearing. Part (i) invokes \"the convergence theorem in [CN]\" to claim that φ̃_j ↘ φ̃ implies φ̃ ∈ E_χ^m(X,ω) and that the weighted energy inequality survives the limit. No such statement is included, and the paper's own Theorem 2.25 is proved for domains and for approximating sequences in E_0^m(Ω,ω), not for the φ̃_j on X. That is exactly where the energy preservation lives; as written it is not verified. The stress-test note has this right. Part (iii) uses truncations φ_s, φ_t that are never defined, and the limit passage again relies on the same unstated convergence theorem. Lemma 3.1's Step 1 also suppresses why the set D is open and why balayage gives H_m(φ̃)=0 there. That one is probably fixable with standard arguments, but it needs to be written down.\n\nNone of this looks fatal. The architecture is sound and the gaps look repairable: state the convergence theorem, verify its hypotheses (capacity bounds for the φ̃_j, or approximation by functions already in E_χ^m(X,ω)), define the truncations, and expand the support argument. The comparison principle and the mass estimates are coherent. I would not desk-reject this; it deserves a serious referee, ideally someone who knows [CN] and CKZ2 and can check whether the quoted black box actually implies the convergence used.\n\nWho gets value: pluripotential theory people working on complex Hessian equations and subextension. The P^n/Lelong-class section is a natural application, though it inherits the same gaps. I would send it to peer review, and I would probably cite the main theorem once the proof is cleaned up, but I would not rely on the current proof of Theorem 3.4.","headline":"A plausible m-subharmonic analogue of CKZ2 that is worth refereeing, but the energy-preservation step in Theorem 3.4 rests on an unstated convergence theorem and needs a rewritten proof.","tokens_in":22070,"tokens_out":1891,"would_cite":true,"duration_ms":19006,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32U15","32Q15","32W20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A quasi-m-hyperconvex domain inside a compact Kähler manifold admits, for every function in the weighted Hessian energy class with bounded mass, a maximal m-subharmonic subextension to the whole manifold that preserves the weighted energy a","keywords":["m-subharmonic functions","maximal subextension","weighted energy class","complex Hessian operator","quasi-m-hyperconvex domain","compact Kähler manifold","Fubini-Study form","m-Lelong class"],"falsifier":"Find a quasi-m-hyperconvex domain Ω ⊂ P^n and φ ∈ E_χ^m(Ω, ω_FS) with ∫_Ω H_m(φ) ≤ ∫_{P^n} ω_FS^n for which the maximal subextension φ̃ satisfies ∫_{P^n} −χ(φ̃)H_m(φ̃) > ∫_Ω −χ(φ)H_m(φ); Theorem 3.4(i) would be false. More directly, exhibit a measure μ on P^n that does not charge m-polar sets, with μ(P^n) = ∫ ω_FS^n, such that the degenerate Hessian equation H_m(u) = μ has no solution u with sup u = −1; this would break the key existence step.","tokens_in":21034,"feed_emoji":"🧮","tokens_out":6278,"duration_ms":52577,"temperature":0.7,"pith_summary":"The paper proves that the subextension problem for m-subharmonic functions—a family interpolating between subharmonic and plurisubharmonic functions—has a positive answer on compact Kähler manifolds. Given a domain Ω with a mild convexity property and a function φ in the weighted energy class whose total Hessian mass does not exceed the total volume of the manifold, there is a maximal m-subharmonic function φ̃ on the whole manifold that lies below φ on Ω, belongs to the same weighted energy class, and does not increase the weighted energy. The construction also controls the complex Hessian measure: inside Ω its trace is no larger than that of φ, and the Hessian measure of φ̃ is carried by the set where φ̃ equals φ together with the boundary of Ω. If correct, this unifies and extends known subextension results from the plurisubharmonic setting to m-subharmonic functions and provides a tool for prescribing complex Hessian measures on projective space.","feed_headline":"Maximal subextensions preserve weighted energy on Kähler manifolds","feed_subtitle":"Extensions to the whole manifold control Hessian mass and touch the original function only on an exact contact set.","key_machinery":"The central object is the complex Hessian operator H_m(u) = (ω+dd^c u)^m ∧ ω^{n−m} acting on ω-m-subharmonic functions, together with the weighted energy class E_χ^m(Ω,ω) defined by finiteness of ∫ −χ(u)H_m(u). The proof is carried by a three-step mechanism: (1) solve, on the whole manifold, degenerate complex Hessian equations H_m(u_j) = 1_Ω H_m(φ_j) + ε_j ω^n with fixed total mass equal to ∫X ω^n, using a known existence theorem for measures that do not charge m-polar sets; (2) a comparison principle forces φ_j ≥ u_j on Ω, so the upper envelopes of the approximating subextensions are well-defined subextensions; (3) monotonicity of the Hessian measure along decreasing sequences (established","core_discovery":"The central claim is Theorem 3.4 (Main Theorem 1.1): if Ω is a quasi-m-hyperconvex domain in a compact Kähler manifold (X,ω) with ∫_Ω ω^n < ∫_X ω^n, χ is a convex weight, and φ ∈ E_χ^m(Ω,ω) satisfies ∫_Ω H_m(φ) ≤ ∫_X ω^n, then the upper envelope φ̃ := sup{ψ ∈ SH_m(X,ω) : ψ ≤ φ on Ω} lies in E_χ^m(X,ω). It satisfies ∫_X −χ(φ̃)H_m(φ̃) ≤ ∫_Ω −χ(φ)H_m(φ), the Hessian control 1_Ω H_m(φ̃) ≤ 1_Ω H_m(φ) as measures, and supp H_m(φ̃) ⊂ {φ̃ = φ} ∪ ∂Ω. The proof approximates φ by bounded functions in E_0^m, uses solutions of degenerate complex Hessian equations on X to build approximating subextensions, and passes to the limit via monotone convergence of Hessian measures.","pith_inferences":["One could test whether the volume inequality ∫_Ω ω^n < ∫_X ω^n can be weakened to ≤, allowing a zero boundary jump; the proof's reliance on the existence theorem might still hold under a matching condition.","The energy inequality could be sharpened: when the volume deficit is zero, one might ask whether ∫_X −χ(φ̃)H_m(φ̃) equals ∫_Ω −χ(φ)H_m(φ), recovering a conservation law for weighted energy.","On P^1 with m=1, the construction should reproduce the known maximal subextension; checking whether the Hessian measure jumps exactly at ∂Ω by the volume deficit would validate the boundary-charge mechanism.","The support containment suggests defining a fine boundary measure of φ with respect to X—the residual measure ν in the decomposition H_m(φ̃) = f·1_Ω H_m(φ) + ν from Remark 3.5—which could be studied as a new invariant of the domain embedding."],"forward_implications":["For any φ in the weighted energy class with total Hessian mass no larger than the manifold's volume, there is a canonical extension φ̃ that is maximal: the largest m-subharmonic minorant of φ on X.","The weighted energy inequality transfers a variational principle from the domain to the ambient manifold: extensions never increase the weighted Hessian energy, so energy minimizers behave well under restriction of the domain.","The support containment means the extension's Hessian measure is zero off the contact set and the boundary; any extra mass appears only as a boundary jump of size ∫X ω^n − ∫Ω H_m(φ), as the paper notes in Remark 3.5.","In the projective case, the correspondence between m-subharmonic functions of logarithmic growth on C^n and ω_FS-m-subharmonic functions on P^n yields global subextensions with prescribed Hessian measure normalized to mass one (Theorem 4.4).","The results provide the m-subharmonic analogue of the classical maximal subextension theory for plurisubharmonic functions, so methods that used that theory can now be attempted for m-subharmonic functions."],"fun_headline_variants":["Maximal subextensions preserve energy, tame Hessian measure","Subextend m-subharmonic functions: energy stays, Hessian bounded","Maximal extension on Kähler: weighted energy intact, Hessian controlled","Energy-preserving maximal subextensions with Hessian cap","Hessian measure controlled in maximal m-subharmonic extensions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof relies on a black-box existence theorem for degenerate complex Hessian equations on compact Kähler manifolds: any measure that does not charge m-polar sets and has the right total mass must be the Hessian measure of some m-subharmonic function normalized to have supremum −1; if that theorem needs extra integrability, the approximation scheme producing the subextension collapses.","fun_headline_variants_meta":{"raw":{"variants":["Maximal subextensions preserve energy, tame Hessian measure","Subextend m-subharmonic functions: energy stays, Hessian bounded","Maximal extension on Kähler: weighted energy intact, Hessian controlled","Energy-preserving maximal subextensions with Hessian cap","Hessian measure controlled in maximal m-subharmonic extensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001482,"raw_usage":{"total_tokens":5808,"prompt_tokens":781,"completion_tokens":5027,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":4938}},"tokens_in":525,"tokens_out":5027,"duration_ms":33256,"temperature":1.0,"reasoning_tokens":4938,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:22:13.566782+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a quasi-m-hyperconvex domain Ω ⊂ P^n and φ ∈ E_χ^m(Ω, ω_FS) with ∫_Ω H_m(φ) ≤ ∫_{P^n} ω_FS^n for which the maximal subextension φ̃ satisfies ∫_{P^n} −χ(φ̃)H_m(φ̃) > ∫_Ω −χ(φ)H_m(φ); Theorem 3.4(i) would be false. More directly, exhibit a measure μ on P^n that does not charge m-polar sets, with μ(P^n) = ∫ ω_FS^n, such that the degenerate Hessian equation H_m(u) = μ has no solution u with sup u = −1; this would break the key existence step.","supporting_citations":[],"review_version":1}