{"id":"2f9a9ec2-951a-418b-a27c-389492c93e76","arxiv_id":"2502.02361","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On Calabi-Yau cones, uniformly bounded cscK metrics are unique up to cone automorphisms, and uniformly bounded Kähler metrics with bounded scalar curvature are asymptotically conical with polynomial decay.","lead":"This paper proves that any constant scalar curvature Kähler metric on a Calabi-Yau cone that is uniformly comparable to the cone metric is the cone metric pulled back by a cone automorphism, and it establishes a Hölder-type estimate showing such metrics approach the cone metric at a polynomial rate near the apex. The results provide a new regularity tool for Kähler metrics with conical singularities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.25's two-sided bound is not shown to imply the homogeneity that Theorem 2.26 uses to conclude [r∂r,V]=0; the Liouville theorem is not proved as written until this step is supplied.","rationale":"The reader's weakest-assumption diagnosis identifies Lemma 2.25 as the key unproved step, and my reading agrees. The central Liouville theorem depends on showing that the scaling vector field V of the tangent cone at infinity commutes with the scaling vector field of the background cone ω_C. That commutator is deduced in Theorem 2.26 from the assertion that V is homogeneous, which in turn is attributed to Lemma 2.25. The proof of Lemma 2.25 is only a sketch and does not give the required spectral analysis for one-forms; the cited Lemma C.1 is for scalar harmonic functions and does not by itself control the coupled one-form system. I do not see an internal contradiction or a likely counterexample, so the appropriate outcome remains a conditional accept pending completion of this step. The other potential concerns (Proposition 2.10's integration argument, the cone-metric conclusion in Theorem 2.21, and technical regularity at the apex) appear more likely to be repairable, whereas Lemma 2.25 is a genuine missing link in the main logical chain. If the spectral argument requested above is supplied and succeeds, no change to the mathematical claims should be needed.","tokens_in":61432,"tokens_out":42656,"duration_ms":509832,"concrete_test":"Complete Lemma 2.25 by expanding a general harmonic one-form α on the cone in an eigenbasis of the Hodge Laplacian on the link L and computing the indicial roots of each mode. Verify explicitly that the two-sided bound r/C ≤ |α|_{ω_C} ≤ Cr forces the norm-growth exponent of every nonzero mode to be exactly 1, kills all logarithmic factors, and leaves only modes whose dual vector field satisfies L_{r∂r}V = 0. If this spectral computation succeeds, Theorem 2.26 follows; if a mode with norm growth exactly r but with L_{r∂r}V ≠ 0 survives, then the step 'V is homogeneous by Lemma 2.25' is false and Corollary 2.35 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.25 (Section 2.3.1) is the hinge between the two-sided linear growth of ψ and the homogeneity of V invoked in Theorem 2.26. Theorem 2.26 states 'as V is homogeneous in r by Lemma 2.25, L_{r∂r}V = μV' and uses μ = 0 to conclude [r∂r,V] = 0. But Lemma 2.25 only asserts that the conclusion of the Hein-Sun classification holds under a two-sided bound; its proof is a sketch saying that a spectral decomposition into one-form modes f_j(x) r^p log^q r forces all terms except the linear-growth no-log term to vanish. That spectral argument is not given. The scalar analogue in Lemma C.1 controls coefficients of scalar eigenfunctions, but the Hodge Laplacian on one-forms is a coupled system, and Lemma 2.24's allowed modes include d(r^μ κ), r^2 η, and r dr, which can in principle mix. The two-sided bound is strong enough that a correct indicial-root argument may well succeed, but as written the paper does not supply it. Since [r∂r,V] = 0 is used to prove that JV is Killing for ω_C (Proposition 2.28), that JV lies in the dual cone C⁰ (Proposition 2.30), and ultimately that the Reeb fields agree (Corollary 2.35), any gap here invalidates the proof of Theorem 2.3 as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on a simply connected Ricci-flat Kähler cone with smooth cross-section: any cscK metric uniformly equivalent to the cone metric is the pullback of that cone metric by a holomorphic automorphism commuting with scaling (Theorem 2.3). It then develops a C^{0,α}-type estimate for Kähler metrics on a ball around the apex with bounded scalar curvature and uniform equivalence to the cone metric (Theorem 3.15), and derives polynomial asymptotics at the apex with rate r^α (Corollary 3.16). The proof combines complex Monge-Ampère theory, heat kernel estimates, tangent cone analysis, classification results for holomorphic vector fields on cones, volume minimization, and a Krylov-type comparison seminorm.","tokens_in":61731,"tokens_out":11021,"duration_ms":105535,"significance":"If the stated results are fully established, the paper would provide a substantial generalization of the classical Liouville theorem for the complex Monge-Ampère equation on C^m to general Calabi-Yau cones, and would give a new route to polynomial asymptotics at conical singularities without assuming smoothability or using Donaldson-Sun theory. The theorem is precisely stated and the overall strategy is coherent, drawing on standard tools in a convincing way. The paper is transparent about its external inputs, and the comparison-set seminorm is a novel technical device. However, several load-bearing steps are only sketched as 'very similar' to earlier results, and the manuscript is not yet in a form where the main theorems can be checked from the text alone.","major_comments":[{"comment":"Lemma 2.25 is the hinge between the two-sided growth bound on the dual one-form ψ and the homogeneity of the scaling vector field V used in Theorem 2.26. The proof is a sketch that refers to a spectral decomposition and says that the argument is 'very similar to Lemma C.1', but Lemma C.1 treats scalar eigenfunctions, whereas Lemma 2.24 concerns the Hodge Laplacian on one-forms, which is a coupled system allowing the modes d(r^μ κ), r^2 η and r dr to mix. The two-sided bound may well imply the desired vanishing of all non-linear-growth terms, but the indicial-root analysis is not supplied. Since Theorem 2.26 obtains [r∂r,V]=0 from this lemma, and this is used in Propositions 2.28 and 2.30 and Corollary 2.35, the missing argument is load-bearing for Theorem 2.3. A complete proof of the one-form version (or a direct proof of L_{r∂r}V=0) is required.","section":"Section 2.3.1, Lemma 2.25"},{"comment":"The iteration argument after equation (2.5) is not justified. The proof selects at the origin a coordinate direction with a positive lower bound on a second derivative of φ∞ and then integrates along that direction 'as long as' the second derivative remains large. It does not control how the coordinate direction changes across iterations: the C^{3,α}_{loc} estimate does not prevent the positive eigenvector of the Hessian from rotating, so the same coordinate direction need not remain available after moving length l_min. Without such control, the claimed linear growth of φ∞ is not established. This step is essential for Corollary 2.11, which concludes ω^m=ω_C^m after rescaling, and hence for the rest of the paper.","section":"Section 2.1, Proposition 2.10"},{"comment":"The proof that the asymptotic limits ω∞ and ω0 are cone metrics rests on the assertion that the Perelman-type functional W(t) is monotone and that the boundary terms in the integration by parts vanish. The text states that the boundary terms 'scale as R^{-3}' at the apex, but no computation is given, and the scaling is not evident from the displayed formulas involving the heat kernel and the distance function. Since the conical structure of ω∞ and ω0 is used to define their Reeb fields and to apply the Martelli-Sparks-Yau classification, this computation is load-bearing. A detailed derivation of dW/dt and the boundary term estimates should be included.","section":"Section 2.2, Theorem 2.21"}],"minor_comments":[{"comment":"There is an empty 'Proof. □' line immediately before the actual proof; remove it.","section":"Section 2.1, Proposition 2.10"},{"comment":"The sentence 'I am highly grateful to my advisor Hans-Joachim Hein for for h is continued support' contains a duplicated 'for' and a typo; it should read 'for his continued support.'","section":"Acknowledgments"},{"comment":"Corollary 2.17 refers to 'Theorem 2.12' for the Gaussian upper bounds, but the relevant statement appears to be Theorem 2.13 (or Definition 2.12); the cross-reference should be corrected.","section":"Section 2.2, Corollary 2.17"},{"comment":"The seminorm in Definition 3.3 depends on a weight function f, but Theorem 3.15 writes [ω]'_{α,B1(o),Σ^2_{3C}×Σ^2_{loc}} without specifying f; state explicitly that the theorem is proved for f=1, or adjust the notation consistently.","section":"Section 3.1, Definition 3.3 and Theorem 3.15"},{"comment":"Proposition 2.8 is stated for Euclidean balls B_6(0) ⊂ C^m, but it is later invoked for balls in a general Calabi-Yau cone (e.g., in Lemma 3.9). Since the cone metric is not Euclidean, either state a local version for a fixed reference metric and record the dependence of constants, or explain the reduction to the Euclidean statement via coordinates.","section":"Section 2.1, Proposition 2.8"}],"recommendation":"major_revision","confidential_remarks":"This is a promising manuscript with important potential results, but in its current form two load-bearing technical points are only sketched: the one-form spectral argument in Lemma 2.25 and the iteration in Proposition 2.10. The entropy computation in Theorem 2.21 is also too compressed to verify. I would encourage the editor to request a major revision rather than reject, because the overall strategy is coherent and the missing arguments appear to be within reach of standard techniques. The paper would benefit from a careful rewriting that replaces 'very similar' statements with complete proofs at the points where the paper's main claims depend on them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves something new and worth taking seriously: a Liouville theorem for cscK metrics on Calabi-Yau cones under uniform equivalence to the cone metric, with no smoothability or strong regularity assumptions. The companion C^{0,α}-type estimate near the apex and the polynomial asymptotics corollary are also new and plausibly useful. The overall strategy is coherent, the citations are honest, and there is no sign of circular reasoning or fitted parameters. This is a serious preprint.\n\nThe soft spots are real, though. The stress-test note is on target: Lemma 2.25 is the hinge for concluding that the scaling vector field V of the tangent cone commutes with r∂r, and its proof is a two-sentence sketch. The scalar eigenfunction argument in Lemma C.1 does not automatically cover the Hodge Laplacian on one-forms, because the system is coupled and the allowed modes in Lemma 2.24 can mix. The two-sided linear-growth bound is strong, and a correct indicial-root argument may well succeed — but it is not supplied. Since [r∂r,V]=0 feeds into Proposition 2.28, then 2.30, and finally Corollary 2.35, the Liouville theorem is not proved as written.\n\nI also share the reader's concern about Proposition 2.10. The iteration along a good coordinate direction assumes the same direction remains good at each step; uniform C^{3,α} control gives a uniform step length, but not that the second derivative in the same fixed direction stays large after moving. That gap is fillable but needs an argument. Theorem 2.21's global cone conclusion is also asserted more quickly than justified; the local cone structure away from the critical point is fine, but the claim that the metric is globally conical needs more care.\n\nNone of this makes me think the main theorems are false. The paper is for Kähler geometers working on complex Monge-Ampère regularity and conical asymptotics; they will get real value from the statements even while reading critically. It deserves a serious referee, and the referee should ask for a written proof of Lemma 2.25, a repaired argument in Proposition 2.10, and a fuller justification of Theorem 2.21.","headline":"A genuinely new Liouville theorem for cscK metrics on Calabi-Yau cones, but the proof as written has a load-bearing gap in Lemma 2.25 that the authors need to fill before the main theorem is established.","tokens_in":62255,"tokens_out":1774,"would_cite":true,"duration_ms":19196,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","53C55","32Q25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any uniformly bounded cscK metric on a Calabi-Yau cone is a pullback of the cone metric by a scaling-compatible automorphism.","keywords":["Calabi-Yau cones","cscK metrics","Liouville theorem","Kähler cones","Reeb fields","Hölder estimates","complex Monge-Ampère equation","tangent cones"],"falsifier":"Test Lemma 2.25 directly: write a non-homogeneous harmonic 1-form on a Calabi-Yau cone with r/C ≤ |α| ≤ Cr, decompose it into eigenfunctions with r^p log^q r factors, and check whether any non-homogeneous term of linear growth survives; if so, the lemma is false. Alternatively, look for a cscK metric uniformly equivalent to the cone metric whose link volume is not the unique volume-minimizing value; Corollary 2.35 would be contradicted.","tokens_in":61230,"feed_emoji":"📐","tokens_out":7185,"duration_ms":62852,"temperature":0.7,"pith_summary":"This paper proves a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on Calabi-Yau cones: any such metric that is uniformly comparable to the cone metric must be a pullback of the cone metric by a holomorphic automorphism commuting with the cone's scaling action. It also proves a Hölder-type ($C^{{0,α}}$) estimate for Kähler metrics on a ball around the cone apex, under only a uniform scalar-curvature bound and uniform equivalence to the cone metric. As a consequence, every such metric is asymptotic to the cone metric with polynomial rate r^α for some small α>0. The significance is that boundedness and constant scalar curvature alone force the metric into the rigid model class of the cone, up to its natural symmetry group, with no smoothability or a priori asymptotic model assumed.","feed_headline":"Bounded cscK metrics on Calabi-Yau cones are rigid","feed_subtitle":"Uniformly bounded constant-scalar-curvature metrics are cone metrics up to a scaling-compatible automorphism.","key_machinery":"The load-bearing mechanism is the blow-up/limit scheme around the apex and infinity. Gaussian heat-kernel estimates and an entropy-type monotonicity show the limits are cone metrics; a classification of harmonic 1-forms with two-sided linear growth forces the scaling vector field of each tangent cone to commute with the scaling field of the original cone and identifies its Reeb field. Volume minimization for Reeb fields and uniqueness of Sasaki-Einstein metrics then identify the tangent cones with the cone metric up to an automorphism in Aut_Scl(C). For the $C^{{0,α}}$ estimate, the central object is a seminorm defined by weighted distance to a comparison set consisting of pullbacks of the cone metric and locally constant (1,1)-forms; blowing up the metric and applying the Liouville theorem gives the estimate.","core_discovery":"The central claim is that the cone metric is the unique cscK metric in its uniform equivalence class, modulo automorphisms that commute with scaling. The proof forces a cscK metric to be Ricci-flat through the complex Monge-Ampère equation and a Liouville argument for bounded harmonic functions, then studies blow-down and blow-up limits, showing they are conical by heat-kernel estimates and an entropy-type monotonicity. The classification of harmonic 1-forms with linear growth is used to show the scaling vector field of any tangent cone commutes with the original scaling field, and volume minimization for Reeb fields together with uniqueness of Sasaki-Einstein metrics shows the tangent cones coincide with the cone metric up to an automorphism. The second result uses a Hölder-style seminorm comparing the metric to pullbacks of the cone metric and to locally constant forms, and a blow-up contradiction that invokes the Liouville theorem to obtain a $C^{{0,α}}$ bound near the apex, then polynomial asymptotics at rate r^α.","pith_inferences":["The same blow-up scheme would likely yield local rigidity for cscK metrics that are only equivalent to the cone metric on compact sets, which could simplify regularity proofs in collapsing or singular limit problems.","The exponent α in the asymptotic rate r^α is probably not sharp; the spectral gap of the Sasaki-Einstein link should determine the optimal decay rate.","The comparison-set seminorm may extend to other conical Kähler models, such as conical Kähler-Einstein metrics with non-zero scalar curvature, giving Hölder estimates at the singular point in those settings.","If the classification lemma breaks down, the first place to look for a counterexample is a harmonic 1-form with non-homogeneous linear growth on a cone; its existence would sever the Reeb-field comparison before volume minimization is invoked."],"forward_implications":["A uniformly bounded cscK metric on a Calabi-Yau cone is automatically Ricci-flat and rigid, up to an automorphism that preserves the scaling vector field.","Uniform equivalence and bounded scalar curvature imply C^{0,α} control near the apex, so the metric has polynomial asymptotics with rate r^α for small α.","The tangent cone at the apex of any metric satisfying these bounds is unique.","The Liouville theorem gives a new route to Evans-Krylov-type estimates for the complex Monge-Ampère equation on Calabi-Yau cones, without assuming smoothability of the cone.","For flat space C^m\\{0}, the proof supplies a new, independent rigidity argument for uniformly bounded cscK metrics, replacing previous approaches that required Ricci-flatness."],"supporting_citations":[{"why":"supplies the classification of homogeneous harmonic 1-forms on cones and of holomorphic vector fields commuting with scaling, which restricts the Reeb field of tangent cones","marker":"[28]"},{"why":"provides the scaling and blow-up framework for collapsing Calabi-Yau metrics that both main proofs are built on","marker":"[29]"},{"why":"gives the volume-minimization principle for Reeb fields used to show tangent cones and the original cone have the same Reeb field","marker":"[35]"},{"why":"supplies the uniqueness of Kähler-Einstein metrics modulo automorphisms, extended in the paper to identify tangent cones with the cone metric","marker":"[4]"},{"why":"extends uniqueness to Sasaki-Einstein metrics, used to conclude the tangent cones are pullbacks of the cone metric by scaling-compatible automorphisms","marker":"[39]"},{"why":"provides the C^{2,α} estimates for complex Monge-Ampère equations used throughout for metric regularity","marker":"[11]"},{"why":"supplies the general Schauder-estimate-by-scaling method that structures the blow-up contradiction arguments","marker":"[46]"},{"why":"defines the classical Hölder-type seminorm whose cone analogue with comparison sets is introduced in the paper","marker":"[32]"},{"why":"provides the Gaussian heat-kernel bounds and Harnack inequality needed to show the asymptotic limits are cone metrics","marker":"[44]"}],"fun_headline_variants":["Calabi-Yau cones: only standard cscK metrics","cscK rigidity on Calabi-Yau cones","Rigid cscK metrics on Calabi-Yau cones","Cone asymptotics from bounded scalar curvature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on a classification of harmonic one-forms on a cone being valid for forms that only satisfy a two-sided linear growth bound, rather than only for exactly homogeneous forms; if that extension fails, the equality of Reeb fields and hence the Liouville theorem are not established.","fun_headline_variants_meta":{"raw":{"variants":["Calabi-Yau cones: only standard cscK metrics","cscK rigidity on Calabi-Yau cones","Rigid cscK metrics on Calabi-Yau cones","Cone asymptotics from bounded scalar curvature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00155,"raw_usage":{"total_tokens":6214,"prompt_tokens":981,"completion_tokens":5233,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":5167}},"tokens_in":597,"tokens_out":5233,"duration_ms":34727,"temperature":1.0,"reasoning_tokens":5167,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T12:26:38.020549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 2.25 directly: write a non-homogeneous harmonic 1-form on a Calabi-Yau cone with r/C ≤ |α| ≤ Cr, decompose it into eigenfunctions with r^p log^q r factors, and check whether any non-homogeneous term of linear growth survives; if so, the lemma is false. Alternatively, look for a cscK metric uniformly equivalent to the cone metric whose link volume is not the unique volume-minimizing value; Corollary 2.35 would be contradicted.","supporting_citations":[{"cited_title":"Calabi-Yau Manifol ds with Isolated Conical Singularities","cited_arxiv_id":null,"evidence_quote":"supplies the classification of homogeneous harmonic 1-forms on cones and of holomorphic vector fields commuting with scaling, which restricts the Reeb field of tangent cones"},{"cited_title":"Higher-O rder Estimates for Collapsing Calabi-Yau Metrics","cited_arxiv_id":null,"evidence_quote":"provides the scaling and blow-up framework for collapsing Calabi-Yau metrics that both main proofs are built on"},{"cited_title":"Sasaki–Einstein Manifolds and Volume Minimisation","cited_arxiv_id":null,"evidence_quote":"gives the volume-minimization principle for Reeb fields used to show tangent cones and the original cone have the same Reeb field"},{"cited_title":"Uniqueness of Einstein Kähler Metrics Modulo Con- nected Group Actions","cited_arxiv_id":null,"evidence_quote":"supplies the uniqueness of Kähler-Einstein metrics modulo automorphisms, extended in the paper to identify tangent cones with the cone metric"},{"cited_title":"Uniqueness of S asaki-Einstein Metrics","cited_arxiv_id":null,"evidence_quote":"extends uniqueness to Sasaki-Einstein metrics, used to conclude the tangent cones are pullbacks of the cone metric by scaling-compatible automorphisms"},{"cited_title":"C2,α -estimate for Monge-Ampère equations with Hölder- continuous right hand side","cited_arxiv_id":null,"evidence_quote":"provides the C^{2,α} estimates for complex Monge-Ampère equations used throughout for metric regularity"},{"cited_title":"Schauder Estimates by Scaling","cited_arxiv_id":null,"evidence_quote":"supplies the general Schauder-estimate-by-scaling method that structures the blow-up contradiction arguments"},{"cited_title":"Lectures on Elliptic and Parabolic Equations in Hölder Spac es","cited_arxiv_id":null,"evidence_quote":"defines the classical Hölder-type seminorm whose cone analogue with comparison sets is introduced in the paper"},{"cited_title":"Uniformly Elliptic Operators on Riemannian Manifolds","cited_arxiv_id":null,"evidence_quote":"provides the Gaussian heat-kernel bounds and Harnack inequality needed to show the asymptotic limits are cone metrics"}],"review_version":1}