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REVIEW 3 major objections 4 minor 1 cited by

Scalar and Mean Curvature Comparison on Compact Cylinder

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that on a compact cylinder with positive scalar curvature and nonnegative boundary mean curvature, a conformally invariant boundary angle condition—the normal tilting less than $\pi/4$ from the interval direction—forces…

desk verdict A plausible and genuinely new extension of Gromov-Lawson/Rade, but the proof is not self-contained and contains a repairable Hessian gap; referee time is justified. read the letter →

arxiv 2507.07005 v1 pith:JU4BFEI3 submitted 2025-07-09 math.DG

Jie Xu This is my paper · ORCID
classification math.DG MSC 53C2153C2035J60
keywords scalarcurvaturemeancompactcylinderpositiveGauss-CodazziequationconformaltransformationellipticPDEanglecondition
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

This paper proves a scalar-curvature comparison theorem for compact cylinders $M=X\times[0,1]$: if the cylinder metric $g$ has positive scalar curvature and nonnegative mean curvature on the boundary, and at the end $X\times\{0\}$ the unit normal $\nu_g$ makes an angle strictly less than $\pi/4$ with the interval direction $\partial_\xi$, then some conformal metric $\tilde g$ on $M$ induces a positive-scalar-curvature metric on $X\cong X\times\{0\}$. The equivalent contrapositive is the sharper statement: when the base $X$ admits no positive-scalar-curvature metric at all, a positive-scalar-curvature cylinder metric obeying the angle condition must have negative mean curvature somewhere on the boundary. This extends a classical torus result to every closed oriented base of dimension at least two, and removes the dimension restrictions of recent partial results, at the cost of the angle hypothesis. The argument is conformal and PDE-based rather than spin-geometric, so it gives a different route into the comparison of scalar and mean curvature on cylinders.

What carries the argument

The load-bearing device is the codimension-two construction: the paper replaces the cylinder $M$ by $M\times S^1_t$ with product metric $\bar g=g\oplus dt^2$, and studies the closed submanifold $W=X\times\{0\}_\xi\times S^1_t$ with induced metric $\sigma^*\bar g$. On $W$ it solves the elliptic equation $4\nabla_V\nabla_V u-4\Delta_{\sigma^*\bar g}u+R_{\bar g}|_W u=F$, where $V$ is the component of $\nu_g$ tangent to the boundary slice after splitting along $\partial_\xi$. The angle condition $g(\partial_\xi,\partial_\xi)/g(\nu_g,\partial_\xi)^2<2$ is exactly what makes the operator $\nabla_V\nabla_V-\Delta_{\sigma^*\bar g}$ elliptic, so the PDE admits a $C^{1,\alpha}$-small solution. Pulling $u_0=u+1$ back to $M\times S^1_t$ gives the conformal factor $(\tilde u')^{4/(n-2)}$; the Gauss-Codazzi equation in its conformal form then converts positivity of $R_{\bar g}$ into positivity of the induced scalar curvature on $X$. The smallness estimates control the error terms in that conversion.

What would settle it

Take $X=T^2$, so $X$ admits no positive-scalar-curvature metric, and construct a sequence of metrics $g_\epsilon$ on $T^2\times[0,1]$ with $R_{g_\epsilon}>0$, $h_{g_\epsilon}\ge0$, and angle tending to $\pi/4$ from below. Numerically solve the auxiliary equation for the conformal factor and check whether the induced scalar curvature on the base stays positive; any example with strict angle inequality, nonnegative mean curvature, and no positive-scalar-curvature metric on the base would refute Corollary 3.1, and a computation at the threshold would show whether the $\pi/4$ bound is sharp.

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Extended reading notes

Core claim

The central claim is Theorem 3.1. Let $X$ be closed and oriented with $\dim X\ge 2$, let $M=X\times[0,1]$, and suppose $g$ has $R_g>0$ on $M$ and $h_g\ge 0$ on $\partial M$. If at $X\times\{0\}$ the metric satisfies $\cos(\angle_g(\nu_g,\partial_\xi))>\sqrt{2}/2$, which is equivalent to $g(\partial_\xi,\partial_\xi)/g(\nu_g,\partial_\xi)^2<2$, then there exists a metric $\tilde g$ conformal to $g$ such that $\iota^*\tilde g$ has positive scalar curvature on $X$. The proof shows that the metric $\tau_1^*\tilde g = (\tilde u')^{4/(n-2)}g$ does the job, where $\tilde u'$ is a positive conformal factor built from the solution of an elliptic PDE on the auxiliary closed manifold $X\times S^1$. The contrapositive, Corollary 3.1, is the paper's advertised comparison: if $X$ carries no positive-scalar-curvature metric and $X\times[0,1]$ carries one satisfying the angle condition on some slice, then the mean curvature must be negative somewhere on the boundary. Corollary 3.2 strengthens the conclusion to a single conformal metric on the cylinder with positive scalar curvature both in the interior and on the induced end metric.

Load-bearing premise

The argument stands on the deferred elliptic-regularity and smallness estimates from the author's earlier work, and on a cited conformal reduction to zero mean curvature; if either fails for the auxiliary circle construction used here, the main theorem would not follow.

Editorial extensions

If this is right

  • For any closed oriented base $X$, deciding whether $X\times[0,1]$ admits a positive-scalar-curvature metric with nonnegative boundary mean curvature reduces, under the angle condition, to whether $X$ itself admits such a metric.
  • The contrapositive gives a boundary obstruction: if $X$ has no positive-scalar-curvature metric, then every positive-scalar-curvature metric on the cylinder satisfying the angle condition must have a boundary point with negative mean curvature.
  • The dimension restriction $\dim X\le 7$ present in earlier partial results is removed whenever the angle condition holds, so the comparison is now proved in all dimensions $n=\dim M\ge 3$.
  • Corollary 3.2 shows one can keep positivity on both the interior and the boundary slice simultaneously within the same conformal class.

Reading between the lines

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

  • The proof is purely conformal and elliptic, so it suggests that the same angle-condition mechanism might settle the analogous comparison for other products $X\times F$ with a transverse vector field, as long as the associated operator is elliptic; the paper does not explore this.
  • The strictness of the angle inequality leaves the threshold case $\cos(\angle)=\sqrt2/2$ open; a natural test is whether the conclusion persists at equality or whether the ellipticity loss is genuine.
  • Because the angle condition is conformally invariant, the theorem is insensitive to conformal rescaling of the cylinder; this may make it easier to combine with existing conformal-volume or Yamabe-type arguments, though the paper does not draw that connection.
  • A concrete extension would be to drop orientability of $X$ or to replace $I$ by a circle $S^1$; the paper assumes a closed oriented base and an interval factor.
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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

3 major / 4 minor

Summary. The paper claims that if a compact cylinder M = X × I, dim M ≥ 3, admits a metric g with positive scalar curvature and nonnegative mean curvature on the boundary, and if the angle between the boundary normal ν_g and ∂_ξ satisfies cos(∠_g(ν_g, ∂_ξ)) > √2/2 on X × {0}, then there exists a metric g̃ on M whose restriction to X × {0} has positive scalar curvature. The contrapositive is presented as a generalization of the Gromov–Lawson torus result: if X admits no positive-scalar-curvature metric and M admits such a metric satisfying the angle condition, then the mean curvature must be negative somewhere. The proof introduces an auxiliary S^1 factor, constructs an elliptic PDE on W = X × S^1, obtains a C^{1,α}-small solution, and uses it as a conformal factor to derive the desired induced scalar curvature via Gauss–Codazzi.

Significance. If the proof were complete, the result would generalize the classical Gromov–Lawson theorem from tori to all closed oriented X with dim X ≥ 2, and would also remove the dimension restrictions in Rade's mean-curvature comparison theorem, under an explicit angle hypothesis. The codimension-two conformal method is a potentially interesting alternative to spin-geometric and μ-bubble techniques. The paper clearly explains the strategy and isolates the angle condition as a conformally invariant hypothesis. However, the present manuscript is not self-contained: the central elliptic estimates are delegated to the author's unpublished preprint [10], and one key Hessian identity in the proof of Theorem 3.1 is false in the stated generality. These issues must be addressed before the main claim can be considered verified.

major comments (3)
  1. [Section 3, proof of Theorem 3.1, after equation (19)] The identity ∇_ν_g ∇_ν_g φ′ = ∇_V ∇_V φ′, justified by 'By (9)', is false for a general cylindrical metric. Writing ν_g = a∂_ξ + V with φ′ independent of ξ, the difference H_{φ′}(ν_g, ν_g) − H_{φ′}(V, V) contains connection terms such as −a²Γ^ℓ_{ξξ}∂_ℓ φ′ − 2aV^i Γ^ℓ_{ξi}∂_ℓ φ′ plus terms from ∂_ξ acting on the coefficients of V; these are generally nonzero unless g is a product metric near X × {0}. The estimates (8) and (10) do not control these second-order terms, and the positivity estimate (20) uses only ∇_V ∇_V u_0 through the PDE (7). The missing Hessian difference must either be estimated and absorbed into the constant C in (17), or the argument must be restructured. As written, the claimed positivity of R_{ı*g̃} is not justified.
  2. [Section 2, Lemmas 2.1, 2.4 and Proposition 2.1] The proof of the main theorem depends crucially on Lemma 2.1 (ellipticity of L′), Proposition 2.1 (existence and C^{1,α}-smallness of the solution of (7)), and Lemma 2.4 (comparison of Laplacians). All three are proved by saying that 'the same argument in [10]' applies, where [10] is the author's own unpublished preprint. In particular, Proposition 2.1 contains the essential improvement ∥u∥_{W^{2,p}} ≤ C∥F∥_{L^p}, which is asserted by analogy rather than proved. Since these estimates are load-bearing for the existence of the conformal factor, the manuscript should either include complete proofs or state and verify precisely which hypotheses of [10] are satisfied by the present W = X × S^1 and operator L. Without this, the central claim cannot be independently checked.
  3. [Section 2, 'Without loss of generality' reduction to h_g ≡ 0] The reduction to zero mean curvature is stated as 'This can be done by taking a conformal transformation of the original metric, see [3]' and is used throughout the proof, including the boundary term in the Yamabe quotient in Lemma 2.3 and the h^2 terms in (19)–(20). It is not obvious that a conformal transformation preserving R_g > 0, the angle condition, and the compact-cylinder structure can always achieve h_g ≡ 0 when the original metric only satisfies h_g ≥ 0. The argument or a precise statement from [3] should be provided, including why the transformed metric still satisfies the hypotheses of Theorem 3.1.
minor comments (4)
  1. [Abstract and Introduction] The text contains multiple typographical errors, including 'metic', 'nonngative', 'repsec', 'transforamtion', 'familiary', and 'bewteen'; a careful proofreading pass is needed.
  2. [Section 2, Lemma 2.4] The notation in Lemma 2.4 is confusing: ∆_{τ*_1 ¯g} u′ is used before the pullback τ*_1 u′ (denoted ˜u′) has been introduced; please clarify the domains and pullbacks in equations (15).
  3. [Section 2, Proposition 2.1] The injectivity step says that the maximum principle and Fredholm alternative give a unique solution, but L is not symmetric because of the ∇_V ∇_V term; please specify the functional-analytic setting and the precise maximum-principle argument for injectivity.
  4. [Section 3, equation (19)] The displayed formula for R_{ı*τ*_1 g̃} is long and contains several terms that are not explicitly identified; adding labels for the terms that are later bounded by C in (20) would improve readability and verifiability.

Circularity Check

3 steps flagged · score 6.0 of 10

The main theorem's proof is not self-contained: the key elliptic smallness lemmas are deferred to a self-cited preprint [10] by the same author, making the central derivation chain dependent on prior unpublished work.

  1. self citation load bearing [Section 2, Lemma 2.1 and its proof]
    "Lemma 2.1. If (6) g(∂ξ, ∂ξ)/g(νg, ∂ξ)^2 < 2, then the operator L′ := ∇V ∇V − ∆σ∗¯g is elliptic on W. Proof. The same argument in [10, Lemma 2.1] follows."

    The ellipticity of the PDE operator is the first load-bearing step in the paper: it is what guarantees a solution u with the small C^{1,α} norm in Proposition 2.1. The proof is not given; it is deferred entirely to [10], an arXiv preprint by S. Rosenberg and J. Xu, which overlaps with the present author. Thus the foundation of the existence argument is imported from the authors' own unpublished work rather than established here.

  2. self citation load bearing [Section 2, Proposition 2.1]
    "Due to the injectivity of the operator, a very similar argument of [10, Proposition 2.1] shows that the Lp estimates can be improved by ∥u∥W 2,p(W,σ∗¯g) ⩽ C∥F ∥Lp(W,σ∗¯g)."

    The crucial smallness estimate ∥u∥C^{1,α}(W) < η, which is later used in Theorem 3.1 to control the error terms in (20), depends on an improved Lp estimate whose proof is not contained in this paper but is referred to the same authors' preprint [10]. Without that estimate, the conformal factor is not known to be small, and the positivity argument in (20) has no quantitative input.

1 more flagged steps
  1. self citation load bearing [Sections 2 and 3, Lemma 2.4 and proof of Theorem 3.1]
    "Lemma 2.4 ... Proof. ... the rest of the proof follows exactly the same as in [10, Lemma 2.4]. ... We point out that the proof of Theorem 3.1 follows closely the argument of [10, Theorem 3.1]."

    The comparison of Laplacians (15), which is used to replace ∆¯gu′ by ∆σ∗¯gu0 in the decisive estimate (20), is not proved here. The theorem's proof is then explicitly described as following the argument of [10, Theorem 3.1], another work by the same author. The central derivation chain therefore rests on a chain of citations to the author's own preprint rather than on a self-contained verification of the geometric reduction.

full rationale

The geometric statement of Theorem 3.1 is not a restatement of its hypotheses: the conclusion that X admits an induced PSC metric is logically distinct from the assumptions Rg>0, hg≥0 and the angle condition, and the paper does attempt to construct a conformal factor through a PDE. However, the construction is not carried out self-containedly. The ellipticity of L′ (Lemma 2.1), the improved Lp estimate giving the small C^{1,α} solution (Proposition 2.1), and the Laplacian comparison (Lemma 2.4) are all justified by 'the same argument' or 'exactly the same as' in [10], a preprint by Rosenberg and Xu with overlapping authorship. These are load-bearing steps: without the smallness of u, the positivity estimate (20) has no quantitative input, and Theorem 3.1 is explicitly said to follow the argument of [10, Theorem 3.1]. Since [10] is neither machine-checked nor independently verified in the present paper, this is self-citation doing load-bearing work rather than external support. A separate correctness concern about the identity ∇ν∇νφ' = ∇V∇Vφ' is not counted as circularity; it is a potential gap, not a definitional equivalence. The score of 6 reflects a central proof that depends on the authors' prior unpublished chain, while the geometric claim still has independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim rests on standard elliptic theory plus a set of technical PDE estimates that are not proved here but referenced to the same author's earlier preprint [10]. No empirical parameters are fitted.

assumptions (5)
  • standard math Elliptic regularity and Sobolev embedding theorems (Agmon-Douglis-Nirenberg)
    Invoked in Proposition 2.1 for Lp estimates and in Lemma 2.3 for H^s local estimates.
  • standard math Maximum principle for elliptic operators on closed manifolds
    Used to show injectivity of the operator L in Proposition 2.1.
  • domain assumption Existence of a conformal transformation making h_g ≡ 0 while preserving R_g > 0
    Reduction at the start of Section 2, cited to Escobar [3]; not proved in the paper.
  • domain assumption Positive relative Yamabe constant for (M×S^1, ∂(M×S^1), [ḡ])
    Used in Lemma 2.3; follows from R_ḡ > 0 and h_ḡ = 0, but the implication is not detailed.
  • ad hoc to paper Validity of the technical lemmas in [10] (Lemma 2.1, Proposition 2.1, Lemma 2.4)
    The core PDE estimates are deferred to the author's own preprint [10]; the paper does not prove them.
invented entities (1)
  • Auxiliary S^1_t space (M × S^1_t, ḡ = g ⊕ dt²)
    purpose: To construct an elliptic PDE whose solution provides the conformal factor; the extra dimension allows a modified second-order operator that is elliptic under the angle condition.
    A mathematical proof device, not a physical postulate; no falsifiable handle outside the construction.

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Cite this review

Pith. "Pith review of Scalar and Mean Curvature Comparison on Compact Cylinder." pith.science (2026). https://pith.science/paper/JU4BFEI3

@misc{pith2026250707005,
  author       = {Pith},
  title        = {Pith review of: Scalar and Mean Curvature Comparison on Compact Cylinder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JU4BFEI3}},
  note         = {Machine review of arXiv:2507.07005}
}
abstract

Let $ X $ be a closed, oriented Riemannian manifold. Denote by $ (M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g) $ a compact cylinder with smooth boundary, $ \dim M \geqslant 3 $. In this article, we address the following question: If $ g $ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $ M $ and nonnegative mean curvature on $ \partial M $; and (ii) the $ g $-angle between normal vector field $ \nu_{g} $ along $ \partial M $ and $ \partial_{\xi} \in \Gamma(TI) $ being less than $ \frac{\pi}{4} $, then there exists a metric $ \tilde{g} $ on $ M $ such that $ \tilde{g} |_{X \times \lbrace 0 \rbrace} $ is a PSC metric on $ X \cong X \times \lbrace 0 \rbrace $. Equivalently, we show that if $ X $ admits no PSC metric, but $ M $ admits a PSC metric $ g $ satisfying the angle condition, then the mean curvature on $ \partial M $ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $ X = \mathbb{T}^{n} $.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Rosenberg $ \mathbb{S}^{1} $-Stability Conjecture for $ \chi(X) = 0 $

    math.DG 2026-07 conditional novelty 7.0 of 10

    For closed oriented manifolds X with dim X ≥ 5 and χ(X)=0, the Rosenberg S¹-stability conjecture holds: X × S¹ admits a PSC metric if and only if X does.

Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [10]

    Rosenberg and J

    S. Rosenberg and J. Xu. A codimension two approach to the S1-stability conjecture. arXiv:2412.12479

    arXiv

  2. [3]

    J. Escobar. The Yamabe problem on manifolds with boundary. J. Differential Geom. , 35:21–84, 1992

    work page 1992

  3. [1]

    Agmon, A

    S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equstions satisfying general boundary conditions I. Commun. Pure Appl. Math , 12:623–727, 1959

    work page 1959

  4. [2]

    Cecchini and R

    S. Cecchini and R. Zeidler. Scalar and mean curvature comparison via the Dirac operator.Geometry and Topology, 28:1167–1212, 2024

    2024

  5. [4]

    M. Gromov. Four lectures on scalar curvature. arXiv:1908.10612v6

    arXiv 1908

  6. [5]

    M. Gromov. Metric inequalities with scalar curvature. Geom. Funct. Anal., 28(3):645–726, 2018

    work page 2018

  7. [6]

    Gromov and B

    M. Gromov and B. Lawson. Spin and scalar curvature in the presence of a fundamental group. I. Ann. of Math. (2), 111(2):209–230, 1980

    work page 1980

  8. [7]

    Kazdan and F

    J. Kazdan and F. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geom., 10:113–134, 1975

    work page 1975

Show all 12 references
  1. [8]

    Lee and T

    J. Lee and T. Parker. The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) , 17(1):37–91, 1987

    1987

  2. [9]

    D. R¨ ade. Scalar and mean curvature comparison viaµ-bubbles. Calc. Var. Partial Differential Equations, 62(187), 2023

    2023

  3. [11]

    J. Xu. Prescribed scalar curvature problem under conformal deformation of a Riemannian metric with Dirichlet boundary condition. arXiv:2208.11318

    arXiv

  4. [12]

    R. Zeidler. Band width estimates via the Dirac operator. J. Differential Geom. , 122(1):155–183, 2022. Department of Mathematics, Northeastern University, Boston, MA, USA Email address : jie.xu@northeastern.edu

    2022

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