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REVIEW 2 major objections 5 minor 28 references

Codimension 2 drawstrings with scalar curvature lower bounds

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.09149 v1 pith:YJSD2NEP submitted 2025-01-15 math.DG

Demetre Kazaras , Kai Xu This is my paper · ORCID
classification math.DG MSC 53C2153C23
keywords drawstringmetricsscalarcurvaturelowerboundscodimension-2submanifoldscollapsinggeometrypulledstringspacesPositiveMassTheoremstabilityLlarull'smovingframemethod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 5.4, proof of Theorem 1.11] 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.
  2. [Abstract and Section 1.3.4] 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.
minor comments (5)
  1. [Theorem 1.5] The statement says the sequence converges 'as i -> 0'; this should read 'as i -> infinity'.
  2. [Theorem 2.4(II)] 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.
  3. [Section 4, proof of Theorem 2.4, Case iii] 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.
  4. [Section 5.3, proof of Theorem 1.7] 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.
  5. [Throughout] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

Direct proof from an explicit ansatz; no circular derivation. Mild self-citation presence only.

full rationale

Walking the derivation chain: Theorem 2.3 derives the pointwise scalar curvature lower bound (2.5) for the explicit ansatz (2.3) from the traced Gauss equations and moving-frame estimates (Theorems 3.7 and 3.12); it does not assume the conclusion. Section 4 constructs h and u explicitly via cutoffs and integral formulas (4.5)-(4.8), with parameters selected by inequalities in Lemmas 4.1 and 4.4, not by fitting the target inequality. The proof of Theorem 2.4 then substitutes the constructed functions into (2.5) and obtains R_{g'} >= R_g - epsilon by dominating each error term; properties (IV)-(VI) are proved from the explicit metric rather than imposed. There is therefore no fitted input renamed as a prediction. The applications (Theorems 1.4, 1.5, 1.6, 1.7, 1.11) invoke Theorem 2.4 with concrete parameter choices together with external convergence results (Basilio-Sormani, Dong-Song); they do not assume their conclusions. The only self-referential aspects are motivational: Section 4 generalizes the earlier [KX23] ansatz, and Section 6 heuristically derives the prototype functions, but Section 6 is explicitly heuristic and is not used in the proof of Theorem 2.3/2.4. The skeptical concern about Section 5.4, namely that the computed 2-form eigenvalues {e^{-2u_i}h_i, 1, h_i} satisfy h_i < 1 in the drawstring region and hence may fail to verify |T|_{g_i} >= |T|_{g_0}, is a correctness/verification question rather than circularity; it does not make the drawstring construction depend on its own output. Overall score 1 reflects the mild presence of self-citation with no significant circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard geometric identities, the tubular neighborhood choice, the specific structural condition on h and u, and several external theorems used as black boxes. The four construction parameters c1, c2, r1, r2 are chosen in the proof to satisfy inequalities, not fitted to data; they parameterize the family of drawstring metrics but are not empirical inputs to the claim.

free parameters (4)
  • c1
    Amplitude of the h(r) perturbation; chosen small in Lemma 4.4 to satisfy the scalar curvature and gluing inequalities; not empirical.
  • c2
    Amplitude of the radial profile w(r) of u; chosen in Lemma 4.4 to satisfy normalization (4.12) and smallness (4.10).
  • r1
    Outer radius of the drawstring region; chosen in Lemma 4.1 depending on epsilon, v0, and the curvature constants C1 to C5.
  • r2
    Inner smoothing radius; chosen in Lemma 4.4 via the continuity of I(r2) so that normalization (4.12) holds.
assumptions (6)
  • standard math Standard Riemannian geometry identities: traced Gauss equation, first variation of mean curvature, Koszul formula, moving frame calculus.
    Used in Section 3, equations (3.1), (3.2), (3.39).
  • domain assumption Tubular neighborhood geometry: for r < 2 r_I the normal exponential map is a diffeomorphism, Sigma_r has mean curvature at least 1/(2r) and area at most 4 pi |Sigma|.
    Stated in Section 2 as a choice of r_I for the setup.
  • domain assumption Condition 2.2: u splits as v(pi(x))w(r) with w(0) = 1, h depends only on r, and the bounds (2.4) hold.
    The scalar curvature estimate (2.5) is proved only for such h and u; the construction in Section 4 verifies these conditions.
  • domain assumption Basilio-Sormani Scrunching Theorem [BS21, Theorem 2.5] used for intrinsic flat convergence to pulled string spaces.
    Used in proofs of Theorems 1.4, 1.5, 1.11, Corollary 1.8, and Proposition A.1.
  • domain assumption Penrose inequality for asymptotically flat 3-manifolds serving as a contradiction in Theorem 1.7.
    Used in Section 5.3 to rule out minimal surfaces in the drawstring example.
  • domain assumption Dong-Song [DS25, Theorem 1.3] used as a black box in Theorem 1.10.
    The application in Section 5.3 relies on their stability result.

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Cite this review

Pith. "Pith review of Codimension 2 drawstrings with scalar curvature lower bounds." pith.science (2026). https://pith.science/paper/YJSD2NEP

@misc{pith2026250109149,
  author       = {Pith},
  title        = {Pith review of: Codimension 2 drawstrings with scalar curvature lower bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJSD2NEP}},
  note         = {Machine review of arXiv:2501.09149}
}
read the original abstract

We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications of this construction are given, primarily on questions concerning the stability of scalar curvature rigidity phenomena, such as Llarull's Theorem and the Positive Mass Theorem.

Figures

Figures reproduced from arXiv: 2501.09149 by the authors.

Figure 1
Figure 1. The cutoff functions that appear when defining [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. The graphs of h and w. Proof. Estimating the cutoff functions by 1 and explicitly integrating, we have ψ(r) ⩽ Z r 0 dρ ρ log2 (1/ρ) + Z min(r,r2) 0 ρ r2 dρ ⩽ 1 log(1/r1) + r2 2 ⩽ 1 2 . Based on the choice of r1 in Lemma 4.1, we determine the remaining parameters c1, c2, r2 according to the following lemma: Lemma 4.4. There exists c1, c2 > 0 and r2 ∈ (0, r1), depending on r1, v0, ε, C1, · · · , C6, R0, such that: (i)… view at source ↗

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