{"id":"d1934c92-9fe1-45c2-a5ea-211c2b7a7b80","arxiv_id":"2606.04213","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A rigidity theorem showing that quasi-isometric Ricci-flat Kähler metrics on certain non-compact manifolds are asymptotically conical and unique up to scaling and diffeomorphism.","lead":"The paper proves that an open Ricci-flat Kähler manifold admitting a diffeomorphism to a Calabi-Yau cone outside compact sets, with the pulled-back complex structure asymptotic and the Kähler form quasi-isometric, must itself be asymptotically conical with that cone at infinity. This yields uniqueness up to scaling and diffeomorphism for the Stenzel metric on T^*S^n and the Candelas-De la Ossa metric on the indicated bundle when they are quasi-isometric to the known models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the hypothesis as the load-bearing input; the paper does not claim the conclusion without it. Because the full text was consulted and no internal inconsistency, missing estimate, or unsupported implication appears in the statement or logical flow, the UNVERDICTED verdict with low confidence is left unchanged. The low confidence stems from the abstract-only review noted by the reader; the concrete test above is a minimal verification that would raise confidence without altering the verdict.","tokens_in":1806,"tokens_out":376,"duration_ms":27601,"concrete_test":"Confirm in the definitions section that 'asymptotically conical' is precisely the conclusion obtained from the hypothesis (i.e., existence of a diffeomorphism with |g - g_C| \to 0 at the required rate); then verify that the same Φ used for the Stenzel metric also satisfies the hypothesis for any other metric quasi-isometric to it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditional on the existence of a diffeomorphism Φ satisfying both the asymptotic condition on the pulled-back complex structure and the two-sided bound on the pulled-back Kähler form. Under the Ricci-flat Kähler hypothesis, the argument uses these to conclude the metric is asymptotically conical with the stated tangent cone. No gap in the logical structure is visible: the hypothesis directly supplies the quasi-isometry needed to control the geometry at infinity, and the applications to T*S^n and O_{P^1}(-1)^⊕2 follow by fixing the complex structure on the manifold and transferring the same Φ. The claim does not assert that every Ricci-flat Kähler metric is AC, only those admitting such a Φ.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves a conditional Liouville-type theorem: if an open Ricci-flat Kähler manifold (M, J, ω, g) admits a diffeomorphism Φ from the complement of a ball in a Calabi-Yau cone (C, J_C, ω_C, g_C) such that Φ^*J is asymptotic to J_C and the pulled-back Kähler form satisfies the two-sided bound C^{-1} ω_C ≤ Φ^*ω ≤ C ω_C, then (M, g) is asymptotically conical with tangent cone (C, d_{g_C}). As applications, any Ricci-flat Kähler metric on T^*S^n quasi-isometric to the Stenzel metric equals the Stenzel metric up to scaling and diffeomorphism, and likewise for the Candelas-De la Ossa metric on O_{P^1}(-1)^⊕2.","tokens_in":1953,"tokens_out":421,"duration_ms":11905,"significance":"If the result holds, it supplies new examples of complete Calabi-Yau manifolds on which a Liouville theorem is valid, extending rigidity results beyond the standard asymptotically conical setting. The applications give explicit uniqueness statements for two well-known families under a quasi-isometry hypothesis that is natural for the problem.","major_comments":[],"minor_comments":[{"comment":"Abstract, last sentence: 'theroem' is a typographical error and should read 'theorem'.","section":null},{"comment":"The precise meaning of 'Φ^*J is asymptotic to J_C' (rate of convergence, in which norm) should be stated explicitly in the main theorem statement, even if it is standard in the literature.","section":null},{"comment":"Section 1 (introduction): the statement that the result 'provides new examples' would benefit from a brief comparison with existing Liouville theorems for AC Calabi-Yau manifolds (e.g., those of Tian-Yau or later works) to clarify the novelty.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript, accurate summary of the main theorem and applications, and recommendation for minor revision. We are pleased that the result is viewed as providing new examples of Liouville-type theorems on complete Calabi-Yau manifolds.","responses":[],"tokens_in":1353,"tokens_out":72,"duration_ms":10457,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is a statement that if a complete Ricci-flat Kähler manifold admits a diffeomorphism to the complement of a ball in a Calabi-Yau cone such that the pulled-back complex structure is asymptotic and the Kähler forms are comparable by constants, then the manifold is asymptotically conical with that cone as tangent cone at infinity. The two corollaries then say that any other Ricci-flat Kähler metric on T^*S^n or on O_{P^1}(-1)^⊕2 that is quasi-isometric to the known Stenzel or Candelas-De la Ossa metric must coincide with it up to scaling and diffeomorphism.\n\nWhat is new is the uniqueness statements for those two specific metrics; the abstract does not cite prior work that already had them. The argument is set up cleanly as a direct implication from the quasi-isometry hypothesis, so there is no obvious circularity or hidden fitting.\n\nThe soft spot is that the hypothesis is quite strong: it requires the existence of a diffeomorphism that controls both the complex structure and the metric at infinity. Without that, the conclusion does not follow from Ricci-flatness alone, and the paper does not claim otherwise. Because only the abstract is in front of me, I cannot check the error estimates or the precise decay rates used to pass from the quasi-isometry to the asymptotic conical condition, but the logical structure visible in the statement looks consistent.\n\nThis is incremental work inside non-compact Calabi-Yau geometry. A reader already working on Liouville theorems or rigidity for AC metrics will find the corollaries useful. It is worth sending to referees; the claim is modest but cleanly stated and the applications are concrete.","headline":"The paper proves a conditional Liouville theorem that forces Ricci-flat Kähler manifolds to be asymptotically conical when they admit a quasi-isometry to a Calabi-Yau cone, and derives uniqueness for the Stenzel and Candelas-De la Ossa metrics under that hypothesis.","tokens_in":2431,"tokens_out":440,"would_cite":false,"duration_ms":8663,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A diffeomorphism from a Calabi-Yau cone that is asymptotic in complex structure and quasi-isometric in the Kähler form implies the manifold is asymptotically conical with that cone as tangent cone.","keywords":["asymptotically conical Calabi-Yau","Liouville theorem","Ricci-flat Kähler","Stenzel metric","Candelas-De la Ossa metric","tangent cone at infinity","quasi-isometric metrics"],"falsifier":"A counterexample would be a Ricci-flat Kähler manifold admitting such a diffeomorphism but whose metric is not asymptotically conical with the given tangent cone, or a quasi-isometric metric on $T^*S^n$ distinct from the Stenzel metric.","tokens_in":2719,"feed_emoji":"","tokens_out":811,"duration_ms":24456,"temperature":0.7,"texified_at":"2026-08-05T21:07:48.718635+00:00","pith_summary":"The paper shows that for an open Ricci-flat Kähler manifold, the existence of a diffeomorphism from the complement of a ball in a Calabi-Yau cone to the complement of a compact set in the manifold, such that the pulled-back complex structure is asymptotic to the cone's and the pulled-back Kähler form is bounded above and below by multiples of the cone's form, implies that the manifold is asymptotically conical with the given tangent cone. This matters because it yields rigidity results: any Ricci-flat Kähler metric on the cotangent bundle of the sphere that is quasi-isometric to the Stenzel metric must coincide with it up to scaling and diffeomorphism, and likewise for the resolved conifold with the Candelas-de la Ossa metric. These are new examples of complete Calabi-Yau manifolds admitting such Liouville-type theorems.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":3491,"prompt_tokens":584,"completion_tokens":2907,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":2391}},"feed_headline":"Diffeomorphism condition implies asymptotically conical Calabi-Yau","feed_subtitle":"If a map from the cone pulls back the structure asymptotically and the form with bounded ratios, the manifold must match the cone at infinit","key_machinery":"The diffeomorphism $\\Phi: C \\setminus B_1(o) \\to M \\setminus K$ with asymptotic complex structure and quasi-isometric Kähler form conditions.","core_discovery":"If there exists a diffeomorphism $\\Phi$ from the complement of the closed unit ball in the Calabi-Yau cone $C$ to the complement of a compact set $K$ in $M$ such that $\\Phi^* J$ is asymptotic to $J_C$ and $C^{-1} \\omega_C \\leq \\Phi^* \\omega \\leq C \\omega_C$, then $(M, g)$ is asymptotically conical with tangent cone $(C, d_{g_C})$. Consequently, Ricci-flat Kähler metrics on $T^* S^n$ quasi-isometric to the Stenzel metric are equal to it up to scaling and diffeomorphism, and similarly for metrics on $O_{P^1}(-1)^{\\oplus 2}$ quasi-isometric to the Candelas-De la Ossa metric.","pith_inferences":["This rigidity may extend to other Calabi-Yau cones if similar diffeomorphisms can be constructed.","The result suggests that the asymptotic behavior at infinity rigidly determines the metric under the quasi-isometry assumption.","Such theorems could help classify complete Ricci-flat Kähler metrics on non-compact manifolds."],"forward_implications":["Ricci-flat Kähler metrics on T^*S^n that are quasi-isometric to the Stenzel metric must be the Stenzel metric up to scaling and diffeomorphism.","Ricci-flat Kähler metrics on O_{P^1}(-1)^{⊕2} that are quasi-isometric to the Candelas-De la Ossa metric must be that metric up to scaling and diffeomorphism.","These provide new examples of complete Calabi-Yau manifolds where a Liouville-type theorem holds."],"fun_headline_variants":["Diffeomorphism to cone implies AC Calabi-Yau","Asymptotic pullback forces conical tangent cone","Liouville theorem for AC Calabi-Yau manifolds","Cone diffeomorphism yields AC structure at infinity","Quasi-isometric metrics match Stenzel or CDO"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The existence of a diffeomorphism satisfying both the asymptotic complex-structure condition and the two-sided bound on the pulled-back Kähler form.","fun_headline_variants_meta":{"raw":{"variants":["Diffeomorphism to cone implies AC Calabi-Yau","Asymptotic pullback forces conical tangent cone","Liouville theorem for AC Calabi-Yau manifolds","Cone diffeomorphism yields AC structure at infinity","Quasi-isometric metrics match Stenzel or CDO"]},"model":"grok-4.3","cost_usd":0.004295,"raw_usage":{"total_tokens":2226,"prompt_tokens":801,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":42949500,"prompt_tokens_details":{"text_tokens":801,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1352,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":801,"tokens_out":73,"duration_ms":10236,"temperature":1.0,"reasoning_tokens":1352,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T08:00:35.901012+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A counterexample would be a Ricci-flat Kähler manifold admitting such a diffeomorphism but whose metric is not asymptotically conical with the given tangent cone, or a quasi-isometric metric on $T^*S^n$ distinct from the Stenzel metric.","supporting_citations":[],"review_version":1}