{"id":"14ac551c-7b2e-4a2f-962a-5c2e69ef4e71","arxiv_id":"2501.09149","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Kazaras and Xu construct drawstring metrics along arbitrary codimension-2 submanifolds with scalar curvature almost bounded below, yielding new collapsed limits and a claimed but flawed Llarull counterexample.","lead":"The paper constructs 'drawstring' metrics that collapse any codimension-2 submanifold to a tiny object while lowering the scalar curvature lower bound by an arbitrarily small amount. These metrics produce new collapsing limits and target stability questions for the Positive Mass Theorem and Llarull's sphere rigidity, though the Llarull application contains an error.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.11's eigenvalue verification fails: the listed 2-form eigenvalues include h_i ≤ 1 (strictly less than 1 for r > 0), so condition (i) of Llarull's theorem is violated, not verified.","rationale":"The reader's verdict (CONDITIONAL) already identifies the Llarull application as problematic, and their rationale explicitly discusses the eigenvalue check in Section 5.4. However, the reader's stated weakest_assumption is about the O(1) constants in the main scalar curvature estimate (Theorem 2.3) and the cancellation in equation (3.53), not about the Llarull application. Examining the central construction, I find no concrete flaw: the scalar curvature estimate (2.5) is derived with explicit error terms, the cancellation of the singular structure-constant derivative in (3.53) is algebraically correct, the constants C1–C5 depend only on the background geometry, and the parameter choices in Section 4 (Lemmas 4.1 and 4.4) are designed to dominate the error terms even for arbitrarily negative v_0. The inequalities relating e^{-2u}|h^{-2}-1| to C e^{2pu}r^{-1}|1-h| are valid using re^{-2nu} ≤ 1 and u ≤ 0. The Llarull issue, by contrast, is a decisive, easily checked failure: the computed eigenvalue h_i is ≤ 1 rather than ≥ 1, so condition (i) of Theorem 1.11 cannot hold. Since this affects only the advertised Llarull instability application and not the central drawstring construction or the other applications, the reader's CONDITIONAL verdict remains appropriate. I do not propose changing the verdict, but I flag that the load-bearing concern is the Llarull eigenvalue error, not the scalar-curvature constants.","tokens_in":48248,"tokens_out":21396,"duration_ms":190817,"concrete_test":"In Section 5.4, compute the g_i -comass norm of the 2-form T = dθ ∧ dt at a point in the drawstring region where r > 0. Since g_i = e^{-2u_i}dr^2 + e^{-2u_i}h_i^2 sin^2 r dθ^2 + e^{2u_i}cos^2 r dt^2, one obtains |T|_{g_i} = h_i sin r cos r = h_i |T|_{g_0}. Theorem 2.4(VII) gives h_i ≤ 1 - r_1 < 1 for sufficiently small r, so |T|_{g_i} < |T|_{g_0}, contradicting condition (i) of Theorem 1.11. Replacing the erroneous check 'eigenvalues > 1 - 1/(100i)' with this exact comparison settles the issue: the claimed counterexample to Llarull stability does not satisfy the Llarull area hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised application to Llarull's theorem (Theorem 1.11) contains a concrete error in Section 5.4. The authors compute the eigenvalues of g_i relative to g_0 on Λ^2 as {e^{-2u_i}h_i, 1, h_i}. By Theorem 2.4(VII), the function h_i satisfies 1 - r_1 ≤ h_i ≤ 1 with r_1 > 0, so h_i < 1 on the drawstring region and h_i = 1 only outside it. The Llarull hypothesis in the introduction (condition (2)) requires |T|_{g_i} ≥ |T|_{g_0} for all 2-forms T, which means every eigenvalue of g_i on 2-forms is at least 1. The eigenvalue h_i is strictly less than 1 in the drawstring region, so the condition fails for the 2-form dθ ∧ dt (whose g_i -norm equals h_i times its g_0-norm). The paper only verifies that the eigenvalues are larger than 1 - 1/(100i), which is a lower bound below 1 and does not imply the required ≥ 1. Consequently, Theorem 1.11 is false as stated. This does not invalidate the central drawstring construction (Theorem 1.1) or the other applications such as collapsing and the positive mass theorem stability examples, but it is a load-bearing overclaim because the abstract and introduction advertise stability of Llarull's theorem as a primary application.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs codimension-2 drawstrings: for any closed oriented codimension-2 submanifold Sigma of an oriented Riemannian n-manifold (M,g), any epsilon>0 and any v0<=0, it produces a metric g' that agrees with g outside a small normal neighborhood, restricts to e^{2v0}g on Sigma, satisfies R_{g'} >= R_g - epsilon, and has small distance from the boundary of the neighborhood to Sigma (Theorems 1.1 and 2.4). The proof combines a moving-frame computation of the scalar curvature of a generalized warped product (Section 3) with an explicit construction of functions h and u (Section 4). The paper then derives several applications: collapsing and partial collapsing of submanifolds, realization of arbitrary conformal distance limits in dimension 3, small-mass asymptotically flat examples without minimal surfaces, a corollary of Dong-Song stability, and a claimed instability of Llarull's theorem for 2-forms.","tokens_in":48446,"tokens_out":14593,"duration_ms":134576,"significance":"If the main construction is correct, it is a substantial technical advance: drawstrings were previously known only for closed geodesics in flat tori, and the present paper extends them to arbitrary codimension-2 submanifolds of arbitrary Riemannian manifolds, with a fully written moving-frame computation and explicit parameter choices. The applications to collapsing, partial collapsing, distance-function limits, and positive-mass stability are meaningful and are argued directly from the construction. The advertised application to Llarull's theorem, however, contains a concrete error in the verification of the 2-form eigenvalue condition, and Theorem 1.11 is false as stated. The core construction and the other applications may still be valid, but the Llarull claim is a load-bearing part of the abstract and introduction and must be repaired or removed.","major_comments":[{"comment":"The verification of condition (i) checks the wrong inequality. The eigenvalues of g_i relative to g_0 on 2-forms are computed as {e^{-2u_i}h_i, 1, h_i}. Theorem 2.4(VII) gives h_i <= 1, with strict inequality on the drawstring region because r_1 > 0 and h = 1 - c_1 eta(r/r_1) psi(r) is strictly less than 1 there. Llarull's condition as stated in the introduction requires |T|_{g_i} >= |T|_{g_0} for all 2-forms T, i.e. every eigenvalue of g_i on Lambda^2 is at least 1. The eigenvalue h_i is strictly below 1, so for T = dtheta wedge dt we have |T|_{g_i} = h_i |T|_{g_0} < |T|_{g_0}. The paper only shows that the eigenvalues are larger than 1 - 1/(100i), which is a bound below 1 and does not imply the required inequality. Consequently Theorem 1.11 is false as stated, and the advertised negative answer to the open question on stability of Llarull's theorem under the 2-form condition is not established.","section":"Section 5.4, proof of Theorem 1.11"},{"comment":"Because the abstract and the introduction list stability of Llarull's theorem as a primary application, the incorrect eigenvalue verification in Section 5.4 is a load-bearing overclaim for the paper's stated contribution. The central drawstring construction and the other applications (Theorems 1.4-1.10) are not affected by this error, but the manuscript's advertised scope is. The authors should either repair the construction or explicitly remove the Llarull claim from the abstract, introduction, and Section 5.4.","section":"Abstract and Section 1.3.4"}],"minor_comments":[{"comment":"The statement says the sequence converges 'as i -> 0'; this should read 'as i -> infinity'.","section":"Theorem 1.5"},{"comment":"The phrase 'we have h(r) = u(r) = 0 for r >= r_1' should read 'h(r) = 1 and u(r) = 0 for r >= r_1', since h is identically 1 outside the drawstring region.","section":"Theorem 2.4(II)"},{"comment":"The displayed inequality in the line following (4.49), namely c_1/(2 r_2 s^4) >= R_g c_1/r, does not follow directly from Lemma 4.1(ii) as written. This is not a fatal issue because, when u <= 0, we have e^{2pu} >= 1 and the preceding inequality (4.46) already gives R_{g'} >= e^{2pu} R_g >= R_g in the case R_g > 0. The authors should replace or simplify the redundant argument.","section":"Section 4, proof of Theorem 2.4, Case iii"},{"comment":"The proof applies Theorem 2.4 to a circle of radius 1/4 and then scales by a factor of 4, but the role of the parameter m and the final ADM mass epsilon should be stated explicitly; as written, the intermediate metric g_{m,epsilon} has mass m outside a ball, and the scaling step is only implicit.","section":"Section 5.3, proof of Theorem 1.7"},{"comment":"There are several typographical errors, including 'summerized', 'devouted', 'apporaches', 'Intrisic-Flat', 'yeilds', and 'Cartesan' coordinates. These do not affect the mathematics but should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core construction in Sections 2-4 appears sound and is a substantial technical contribution. The failure of Theorem 1.11 is localized to Section 5.4 and the surrounding presentation, but it is advertised as a primary application, so the manuscript currently overclaims. I recommend major revision rather than rejection because the main drawstring theorem and the other applications seem defensible; the authors should repair or remove the Llarull claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious geometry paper, and the main drawstring construction is probably the real thing. The Llarull application in Theorem 1.11 is not. I agree with the stress-test note.\n\nWhat is new: KX23 built drawstrings around geodesics in 3-tori; LNN23 needed codimension at least 3. Here they handle any oriented codimension-2 submanifold in any dimension n >= 3, with prescribed conformal factor on Sigma, and they prove the scalar curvature lower bound directly from the ansatz. The applications to collapsing, partial collapsing, conformal limits, and the positive mass theorem are natural and mostly follow from the main theorem. The moving-frame computation is detailed, and Section 4 makes the parameter choices explicit. This is worked-out mathematics, not a sketch.\n\nThe soft spot I would insist on is Theorem 1.11. In Section 5.4 the eigenvalues of g_i relative to g_0 on 2-forms are listed as {e^{-2u_i}h_i, 1, h_i}. Llarull's hypothesis requires every such eigenvalue to be at least 1. By the paper's own Theorem 2.4(VII), h_i <= 1 with strict inequality inside the drawstring region, so the eigenvalue h_i is strictly less than 1 on the form dtheta ^ dt and condition (i) fails. There is also an inconsistency in the same paragraph: the stated vector eigenvalues {e^{-u}, e^{-u}h, e^u} do not match the metric expression (2.3) with p=1, which gives {e^{-2u}, e^{-2u}h^2, e^{2u}}. Either way, the 2-form Llarull condition is not satisfied, and the counterexample claim in the abstract and introduction is unsupported.\n\nIs this fatal to the rest? No. The main estimate Theorem 2.3 and the drawstring construction are independent of the Llarull application. The collapse examples and the positive mass theorem applications look internally consistent, and the reader's conditional verdict is about right. A referee should ask the authors to correct or delete Theorem 1.11 and fix the eigenvalue statement before publication; the main result could survive that revision.\n\nCitation pattern: the reliance on the authors' own KX23 is natural, and the other references are appropriate. No red flags there.\n\nWho this is for: people working on scalar curvature stability, Gromov-Hausdorff and intrinsic flat limits, and Llarull-type rigidity. It deserves a serious referee, but the referee should be told to examine the eigenvalue calculation in Section 5.4 first.","headline":"The drawstring construction looks like a real step forward, but the advertised Llarull counterexample has a sign error in the eigenvalue check and should not be accepted as stated.","tokens_in":49120,"tokens_out":4449,"would_cite":true,"duration_ms":46068,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","53C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a drawstring—a metric modification that collapses a submanifold's intrinsic diameter to epsilon at scalar-curvature cost epsilon—can be placed along any oriented codimension-2 submanifold, with any prescribed…","keywords":["drawstring metrics","scalar curvature lower bounds","codimension-2 submanifolds","collapsing geometry","pulled string spaces","Positive Mass Theorem stability","Llarull's Theorem","moving frame method"],"falsifier":"Take M = $R^{3}$ with the Euclidean metric, Sigma the unit circle, v0 = log epsilon, and numerically evaluate R_{g'} for the functions h,u constructed in (4.5)-(4.8) with the paper's parameter choices; if any point has R_{g'} < -epsilon (or R_{g'} < 0 in the asymptotically flat example of Theorem 1.7), the claim that the error terms are dominated by c1/($r^{2}$ $log^{3}$(1/r)) is false. Alternatively, directly check the key identity (3.53) for the frame from Theorem 3.1: if the e_a gamma_{a $\\theta$ $\\theta$} term fails to cancel the singular contribution, the scalar curvature acquires an unbounded $r^{{-1}}$ term near Sigma.","tokens_in":47909,"feed_emoji":"🪢","tokens_out":5937,"duration_ms":56216,"temperature":0.7,"pith_summary":"The paper establishes that a scalar-curvature drawstring can be created along any oriented codimension-2 submanifold: given any epsilon > 0 and any nonpositive conformal factor v0, there is a nearby metric g' that agrees with g outside a tiny tube, restricts to $e^{{2v0}}$g on the submanifold, and satisfies R_{g'} >= R_g - epsilon everywhere. This means a prescribed submanifold can be turned into a shortcut—its intrinsic diameter becomes arbitrarily small—while the scalar curvature lower bound is barely disturbed. The result matters because stability questions for scalar curvature rigidity, such as the Positive Mass Theorem and Llarull's Theorem, ask which collapsing behaviors are possible under almost nonnegative scalar curvature. The paper uses the construction to produce pulled-string limits, arbitrary conformal distance limits in dimension 3, and counterexamples to Intrinsic-Flat stability conjectures. If correct, codimension-2 collapsing is a general phenomenon rather than a special feature of curves.","feed_headline":"Drawstrings can collapse any codimension-2 submanifold","feed_subtitle":"New metrics put a shortcut on any prescribed slice, lowering scalar curvature by less than epsilon.","key_machinery":"The central machinery is the warped drawstring ansatz g' = $e^{{-2(n-2)u}}$$dr^{2}$ + $e^{{-2(n-2)u}}$$h^{2}$ $omega_theta^{2}$ + $e^{{2u}}$g_H in a tube around Sigma. The functions h and u are built from the prototypes h(r) = 1 - c1 eta(r/r1)psi(r) and u(x) = v0(pi(x))w(r), with w(0)=1, so that the fiber metric $dr^{2}$ + $h^{2}$ $omega_theta^{2}$ is a smoothed acute two-dimensional cone. The scalar curvature is computed by the method of moving frames, and the load-bearing algebraic step is the cancellation of the singular term e_theta gamma_theta aa in equation (3.53), which removes an apparent $r^{{-1}}$ contribution to R_{g'}. The remaining error terms are controlled by the frame estimates of Theorem 3.1 and Lemma 3.11, and the parameter choices in Section 4 ensure that the positive term c1/($r^{2}$ $log^{3}$(1/r)) dominates the five error constants C1 through C5, yielding R_{g'} >= R_g - epsilon.","core_discovery":"The central claim is Theorem 1.1: for any closed oriented embedded codimension-2 submanifold Sigma of an oriented Riemannian n-manifold (M,g), any epsilon > 0, and any v0 in C^infinity(Sigma) with v0 <= 0, there is a metric g' with g' = g outside N_g(Sigma, r1), g'|Sigma = $e^{{2v0}}$g|Sigma, R_{g'} >= R_g - epsilon everywhere, and d_{g'}(x, Sigma) <= 3r1 for x in the boundary of N_g(Sigma, r1). The metric is written in Fermi-type coordinates as g' = $e^{{-2(n-2)u}}$$dr^{2}$ + $e^{{-2(n-2)u}}$$h^{2}$ $omega_theta^{2}$ + $e^{{2u}}$g_H, where omega_theta is the unit angular form in the two-dimensional normal bundle and H is the horizontal distribution. The proof chooses h and u so that near Sigma the metric resembles a smoothed acute two-dimensional cone, whose tip generates large positive scalar curvature that pays for the conformal degeneration g'|Sigma = $e^{{2v0}}$g|Sigma and for the gluing errors. The main technical estimate, Theorem 2.3, bounds R_{g'} below by $e^{{2pu}}$R_g plus terms involving h,u minus five error terms, and the construction of h,u in Section 4 makes the positive cone term dominate all errors.","pith_inferences":["One natural extension is to nonorientable codimension-2 submanifolds: the moving-frame estimates are local, so passing to the orientation double cover of the normal bundle may remove the orientability hypothesis.","The same cone-smoothing mechanism may place drawstrings along higher-codimension submanifolds more easily, since higher-dimensional normal bundles give even more positive cone curvature; the paper does not pursue this.","Theorem 1.6's uniform distance convergence suggests that scalar curvature lower bounds are not closed under pointwise convergence of distance functions in dimension 3, sharpening the higher-dimensional examples of Lee-Topping.","The asymptotically flat examples of Theorem 1.7 behave like wormhole shortcuts without apparent horizons; a natural test is whether a spacetime extension of these metrics preserves that no-horizon property."],"forward_implications":["Drawstrings can be placed along any oriented codimension-2 submanifold, so any compact connected submanifold of arbitrary codimension can be collapsed to a point by a sequence of metrics with R_{g_i} >= R_g - 1/i.","The distance functions converge to c-partially pulled string metrics, giving Gromov-Hausdorff limits that are pulled string spaces; for n = 3 and c = infinity the convergence is also Intrinsic-Flat.","In dimension 3, positive scalar curvature is not preserved under uniform convergence of distance functions: arbitrary metrics in a Yamabe-positive conformal class can be realized as distance limits of metrics with R > lambda.","There exist asymptotically flat 3-manifolds with R >= 0, arbitrarily small ADM mass, no closed minimal surfaces, and a short loop, giving a counterexample to Intrinsic-Flat stability conjectures for the Positive Mass Theorem.","There exist 3-spheres satisfying the 2-form area condition |T|_{g_i} >= |T|_{g0}, R >= 6 - 1/i, and uniformly positive Cheeger constants that still converge to a pulled string space, so the 2-form version of Llarull's Theorem has no such stability."],"supporting_citations":[{"why":"Introduced drawstrings along closed geodesics and the prototype functions h = 1 - c1/log(1/r) and u = -c2 log log(1/r) that this paper generalizes.","marker":"[KX23]"},{"why":"Supplies the moving-frame method used to compute scalar curvature with a non-integrable horizontal distribution.","marker":"[Zho23]"},{"why":"Constructed earlier drawstring-like examples collapsing geodesics in dimensions n >= 4 using codimension at least 3 cone positivity, the baseline this paper improves.","marker":"[LNN23]"},{"why":"Defines pulled string spaces and the sewing techniques used in the Intrinsic-Flat convergence arguments.","marker":"[BDS18]"},{"why":"Provides the Scrunching Theorem used to obtain Intrinsic-Flat convergence to pulled string spaces in dimension 3.","marker":"[BS21]"},{"why":"The Dong-Song bad-set stability result whose sharpness is probed by Theorem 1.10 and Corollary 1.8.","marker":"[DS25]"},{"why":"The all-dimensions Llarull stability theorem whose no-bubble hypothesis is shown necessary by Theorem 1.11.","marker":"[HZ24]"},{"why":"The dimension-three stability result for Llarull's theorem that motivated the 2-form instability question.","marker":"[ABK23]"}],"fun_headline_variants":["Codim-2 collapse costs less than epsilon scalar curvature","New metrics collapse any codim-2 submanifold with curvature control","Prescribed shrink on codim-2 slices, scalar curvature drops by epsilon","Cone construction collapses codim-2 sets, preserves scalar curvature bounds","Stability of scalar curvature rigidity survives codim-2 collapse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the scalar-curvature estimate (2.5): the moving-frame constants in Lemma 3.11 and the cancellation in (3.53) must be strong enough that the positive cone term c1/($r^{2}$ $log^{3}$(1/r)) dominates the five error terms C1 through C5 for all r < r1, and if any of those constants were larger than the paper's bounds the inequality R_{g'} >= R_g - epsilon could fail.","fun_headline_variants_meta":{"raw":{"variants":["Codim-2 collapse costs less than epsilon scalar curvature","New metrics collapse any codim-2 submanifold with curvature control","Prescribed shrink on codim-2 slices, scalar curvature drops by epsilon","Cone construction collapses codim-2 sets, preserves scalar curvature bounds","Stability of scalar curvature rigidity survives codim-2 collapse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1716,"prompt_tokens":879,"completion_tokens":837,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":744}},"tokens_in":495,"tokens_out":837,"duration_ms":8609,"temperature":1.0,"reasoning_tokens":744,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:15:14.125781+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take M = $R^{3}$ with the Euclidean metric, Sigma the unit circle, v0 = log epsilon, and numerically evaluate R_{g'} for the functions h,u constructed in (4.5)-(4.8) with the paper's parameter choices; if any point has R_{g'} < -epsilon (or R_{g'} < 0 in the asymptotically flat example of Theorem 1.7), the claim that the error terms are dominated by c1/($r^{2}$ $log^{3}$(1/r)) is false. Alternatively, directly check the key identity (3.53) for the frame from Theorem 3.1: if the e_a gamma_{a $\\theta$ $\\theta$} term fails to cancel the singular contribution, the scalar curvature acquires an unbounded $r^{{-1}}$ term near Sigma.","supporting_citations":[],"review_version":1}