{"id":"e5cb76c2-11d7-4a27-8650-922294187f3b","arxiv_id":"2606.28767","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs infinite-time singularities for Lagrangian mean curvature flow in Gibbons-Hawking spaces where H converges uniformly to 0 but log max |A| ~ sqrt(t) as t to infinity.","lead":"The paper constructs specific circle-invariant Lagrangian 2-spheres in Gibbons-Hawking spaces whose mean curvature flow exists for all time, converges to a chain of special Lagrangian spheres, has mean curvature going uniformly to zero, yet has second fundamental form blowing up at rate log max |A| comparable to sqrt(t). A smart generalist might read it to see how infinite-time singularities can form even when mean curvature vanishes, refining prior convergence pictures in ge","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Barrier family construction assumes initial C^2-closeness and concavity are preserved long enough for trapping to control all-time asymptotics; this is not independently verified beyond the initial data hypothesis.","rationale":"The reader's weakest_assumption directly identifies the barrier-trapping step as load-bearing. Because the full proof is described as 'based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics,' any gap in justifying that the initial closeness persists under the flow immediately undermines both the infinite-time existence and the sqrt(t) rate. This is an internal dependence rather than an external consensus issue, so the verdict moves from UNVERDICTED to CONDITIONAL pending verification that the assumed regime is invariant.","tokens_in":1732,"tokens_out":462,"duration_ms":19074,"concrete_test":"Extract the explicit ODE or evolution equation satisfied by the quotient curves (likely in §3 or §4). Numerically integrate the curve-shortening-type flow starting from a concave C^2-close initial segment (e.g., a small perturbation of n collinear intervals) for time up to T=100; check whether concavity is lost or the C^2-distance to the collinear configuration exceeds the barrier width before the mean curvature drops below 10^{-3}. If either occurs, the trapping hypothesis is violated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (infinite-time smooth existence + convergence to A_{n-1} chain with log max |A| ~ sqrt(t)) rests on a one-parameter family of barrier curves that trap the quotient curves. The abstract and reader's weakest_assumption both state that this trapping requires the initial quotient curves to be concave and C^2-close to consecutive collinear segments. If the flow causes the curves to lose concavity or drift outside the C^2-neighborhood before the barriers can enforce the claimed decay of mean curvature and growth of |A|, the trapping argument fails and the rate estimate does not follow. The paper provides no separate maximum-principle or evolution equation showing that concavity/closeness is preserved under the Lagrangian MCF in Gibbons-Hawking space; the barrier analysis is therefore the sole support for both existence and the precise blow-up rate.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper constructs infinite-time singularities for Lagrangian mean curvature flow of circle-invariant Lagrangian 2-spheres in Gibbons-Hawking spaces. For initial quotient curves that are concave and C²-close to consecutive collinear segments, it claims the flow exists smoothly for all time, converges to an A_{n-1}-chain of special Lagrangian spheres, with mean curvature converging uniformly to zero while log max |A| grows like √t. The proof relies on a one-parameter family of barrier curves whose asymptotics trap the evolving curves.","tokens_in":1936,"tokens_out":568,"duration_ms":23709,"significance":"If correct, the result refines the infinite-time convergence picture of Lotay-Oliveira by establishing curvature blow-up (rather than bounded curvature) together with a precise rate in this semi-stable setting. The barrier-curve method supplies an explicit construction and asymptotic control that could serve as a template for related singularity analyses in Lagrangian MCF.","major_comments":[{"comment":"The central trapping argument (abstract, paragraph on construction; also the barrier-family analysis) requires that the initial concavity and C²-closeness to collinear segments be preserved for all time so that the one-parameter family of barriers can continue to enclose the quotient curves. No evolution equation, maximum principle, or separate a-priori estimate is supplied to establish preservation of these properties under the Lagrangian MCF in the Gibbons-Hawking metric; without it the trapping and the resulting rate log max |A| ∼ √t rest on an unverified hypothesis.","section":"barrier construction / proof outline"},{"comment":"The claimed uniform convergence of mean curvature to zero while |A| blows up is derived from the asymptotics of the barrier curves. The manuscript must therefore verify that the error between the actual quotient curve and the barriers remains small enough throughout the infinite-time regime to justify the √t growth; the current sketch does not display the requisite error estimates or comparison principle that would close this gap.","section":"asymptotics of barrier curves"}],"minor_comments":[{"comment":"Notation for the Gibbons-Hawking metric and the circle action should be introduced with explicit coordinate expressions early in the paper to make the reduction to quotient curves self-contained.","section":"introduction / setup"},{"comment":"The statement that the flow 'converges to the associated A_{n-1}-chain' would benefit from a precise definition of the topology or distance in which convergence holds (e.g., in C^∞ on compact sets away from the singular loci).","section":"main theorem statement"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"Dear Editor,\n\nWe thank the referee for their careful reading and for identifying points where the manuscript requires additional detail to make the arguments fully rigorous. We address each major comment below and will incorporate the necessary clarifications and estimates in a revised version.","responses":[{"response":"We agree that explicit verification of the preservation of concavity and C²-closeness is essential for the trapping argument to hold for all time. The current manuscript sketch relies on these properties without deriving the necessary evolution equations or applying the maximum principle. In the revision we will add a dedicated subsection that computes the evolution of the relevant quantities (signed curvature and deviation from collinearity) under the Lagrangian MCF in the Gibbons-Hawking metric and shows, via the parabolic maximum principle, that the initial concavity and C²-closeness are preserved as long as the flow remains smooth. This will close the gap and justify continued use of the barrier family.","revision_made":"yes","referee_comment":"The central trapping argument (abstract, paragraph on construction; also the barrier-family analysis) requires that the initial concavity and C²-closeness to collinear segments be preserved for all time so that the one-parameter family of barriers can continue to enclose the quotient curves. No evolution equation, maximum principle, or separate a-priori estimate is supplied to establish preservation of these properties under the Lagrangian MCF in the Gibbons-Hawking metric; without it the trapping and the resulting rate log max |A| ∼ √t rest on an unverified hypothesis."},{"response":"We concur that quantitative control on the distance between the evolving quotient curve and the barrier family is needed to transfer the barrier asymptotics to the actual solution and obtain the precise √t rate. The manuscript currently provides only a qualitative trapping statement. In the revision we will insert a comparison lemma that establishes a uniform-in-time bound on the C²-distance between the solution and the nearest barrier curve, together with an error estimate that remains o(1) relative to the barrier separation as t → ∞. This will rigorously justify both the uniform convergence of mean curvature to zero and the claimed growth rate of log max |A|.","revision_made":"yes","referee_comment":"The claimed uniform convergence of mean curvature to zero while |A| blows up is derived from the asymptotics of the barrier curves. The manuscript must therefore verify that the error between the actual quotient curve and the barriers remains small enough throughout the infinite-time regime to justify the √t growth; the current sketch does not display the requisite error estimates or comparison principle that would close this gap."}],"tokens_in":1387,"tokens_out":561,"duration_ms":25542,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors give an explicit construction of infinite-time singularities for Lagrangian mean curvature flow in Gibbons-Hawking spaces. The flow exists forever, mean curvature goes uniformly to zero, but the second fundamental form blows up with log of its max norm growing like sqrt(t). This adds a blow-up rate to the convergence result of Lotay and Oliveira.\n\nThey work with circle-invariant Lagrangian spheres whose quotient curves start concave and C2-close to collinear segments. A one-parameter family of barrier curves traps the evolving curves and lets them analyze the long-time behavior, showing convergence to an A_{n-1} chain of special Lagrangian spheres.\n\nThe barrier approach and the asymptotic estimates look like the new technical contribution. It gives a concrete model in this semi-stable setting.\n\nThe soft spot is the preservation of the initial assumptions. The trapping depends on the curves staying concave and sufficiently close for all time. The paper does not appear to provide a separate argument that the flow preserves concavity or the C2 neighborhood, so the barriers have to do all the work. If that holds up in the details it is fine, but it is the load-bearing part.\n\nThis paper is for people already working on Lagrangian mean curvature flow or special Lagrangians in hyperkähler 4-manifolds. It is too specialized for a broad audience, but the explicit rate makes it useful for that group.\n\nIt deserves serious refereeing because the statement is precise and the method can be checked.","headline":"This paper adds an explicit blow-up rate to an infinite-time convergence result for Lagrangian MCF in a special geometric setting.","tokens_in":2417,"tokens_out":374,"would_cite":false,"duration_ms":34492,"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":"Lagrangian mean curvature flow in Gibbons-Hawking spaces develops infinite-time singularities where mean curvature vanishes uniformly but the second fundamental form blows up with log max |A| scaling as sqrt(t).","keywords":["Lagrangian mean curvature flow","Gibbons-Hawking spaces","infinite-time singularities","vanishing mean curvature","curvature blow-up","special Lagrangian spheres","barrier curves"],"falsifier":"A direct computation or numerical simulation of the flow from such initial data in which max |A| remains bounded for all time, or in which log max |A| grows at a rate other than sqrt(t), would show the claimed blow-up does not occur.","tokens_in":2602,"feed_emoji":"","tokens_out":696,"duration_ms":22118,"temperature":0.7,"pith_summary":"The paper constructs circle-invariant Lagrangian 2-spheres in Gibbons-Hawking spaces whose quotient curves are concave and C2-close to consecutive collinear segments. Starting from this data, the associated Lagrangian mean curvature flow exists smoothly for all time and converges to an A_{n-1}-chain of special Lagrangian spheres. Mean curvature converges uniformly to zero in the limit, yet the second fundamental form becomes unbounded, with the precise rate log max |A(·,t)| comparable to sqrt(t) as t tends to infinity. The argument relies on constructing a one-parameter family of barrier curves whose asymptotics trap the flow and produce the claimed behavior. This refines earlier infinite-time convergence results by establishing curvature blow-up in the semi-stable regime.","feed_headline":"Lagrangian flow converges to special spheres but curvature blows up","feed_subtitle":"Mean curvature vanishes uniformly while log of max second fundamental form grows like sqrt(t) in Gibbons-Hawking spaces","key_machinery":"One-parameter family of barrier curves for the quotient curves that trap the evolving curve and control its long-time asymptotics.","core_discovery":"We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons-Hawking spaces. We consider circle-invariant Lagrangian 2-spheres whose quotient curves are concave and are C2-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated A_{n-1}-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, log max |A(·,t)| is comparable to sqrt(t) as t to infinity. The proof is based on a one-parameter family of barrier curves and a detailed analysis of th","pith_inferences":["The barrier-curve method may extend to other invariant Lagrangian mean curvature flows without requiring full circle symmetry.","The sqrt(t) growth rate for log |A| suggests a possible scaling law that could be tested in related parabolic flows with vanishing mean curvature.","Numerical integration of the quotient-curve evolution under the same concavity and closeness assumptions could independently confirm the blow-up rate."],"forward_implications":["The flow converges to the A_{n-1}-chain despite unbounded curvature.","Infinite-time singularities with vanishing mean curvature are realized explicitly in this setting.","The blow-up rate satisfies log max |A(·,t)| ~ sqrt(t).","The construction refines the infinite-time convergence picture for the semi-stable case."],"fun_headline_variants":["Curvature blowup in infinite-time Lagrangian singularities with zero mean curvature","Lagrangian mean curvature flow converges but curvature unbounded at infinite time","Gibbons-Hawking Lagrangian flow shows sqrt(t) growth in log max second fundamental form","Infinite-time convergence with curvature blowup in Gibbons-Hawking spaces"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The initial quotient curves must be concave and C2-close to consecutive collinear segments so the one-parameter family of barrier curves can enclose and trap the flow.","fun_headline_variants_meta":{"raw":{"variants":["Curvature blowup in infinite-time Lagrangian singularities with zero mean curvature","Lagrangian mean curvature flow converges but curvature unbounded at infinite time","Gibbons-Hawking Lagrangian flow shows sqrt(t) growth in log max second fundamental form","Infinite-time convergence with curvature blowup in Gibbons-Hawking spaces"]},"model":"grok-4.3","cost_usd":0.010027,"raw_usage":{"total_tokens":4457,"prompt_tokens":677,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":100274500,"prompt_tokens_details":{"text_tokens":677,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3702,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":677,"tokens_out":78,"duration_ms":32000,"temperature":1.0,"reasoning_tokens":3702,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T09:01:32.506032+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct computation or numerical simulation of the flow from such initial data in which max |A| remains bounded for all time, or in which log max |A| grows at a rate other than sqrt(t), would show the claimed blow-up does not occur.","supporting_citations":[],"review_version":1}