Fill-Ins of Tori with Scalar Curvature Bounded from Below

Yipeng Wang

REVIEW 2 major objections 4 minor 25 references

Fill-Ins of Tori with Scalar Curvature Bounded from Below

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For flat tori $S^1 \times T^{n-2}$ with $3 \le n \le 7$, fill-ins with scalar curvature $\ge -n(n-1)$ and positive mean curvature obey a sharp total-mean-curvature bound whose constant is attained in the limit.

desk verdict The flat-torus sharp estimate in Theorem 1.6 is likely correct and worth refereeing, but the advertised general torus-boundary result (Theorem 1.8) is not proven as written because the final rescaling step in the proof of Theorem 1.7 changes the prescribed boundary metric. read the letter →

arxiv 2411.14667 v1 pith:7VO52P7P submitted 2024-11-22 math.DG

Yipeng Wang This is my paper · ORCID

classification math.DG MSC 53C2053C2153C24

keywords scalarcurvaturetotalmeanfill-inflattorussystolicinequalityprescribedquasi-localmassconjecture

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 establishes a sharp upper bound on the total mean curvature of the boundary of a Riemannian fill-in of a flat torus $\Sigma = S^1 \times T^{n-2}$ with $3 \le n \le 7$, assuming the interior scalar curvature is at least $-n(n-1)$ and the boundary mean curvature is positive. The bound depends only on the dimension and the flat boundary metric, through the length of the shortest non-contractible circle in the $S^1$ factor. The result settles a special case of a conjecture about total mean curvature of fill-ins with scalar curvature bounded from below. The proof combines the prescribed scalar curvature equation with a recently proved systolic inequality for tori.

What carries the argument

The central object is the prescribed scalar curvature equation on the hyperbolic cylinder $\Sigma \times [\rho_0, \infty)$ with metric $g = \rho^2\gamma + u^2\rho^{-2}d\rho^2$; the scalar-curvature condition $R_g = -n(n-1)$ reduces to $(n-1)\rho\,\partial_\rho u = u^2\Delta_{\Sigma_\rho}u - \frac{1}{2}n(n-1)(u^3-u)$. Solving this equation with boundary data coming from the fill-in yields the asymptotic expansion $u = 1 + \mu(x)\rho^{-n} + O(\rho^{-n-2})$. A perturbation of the level sets of $\rho$ by a function that solves a Laplacian equation adjusts the mean curvature of a large torus slice to the desired constant. A gluing lemma attaches this cylinder to the fill-in while nearly preserving the lower scalar-curvature bound, and a systolic inequality on the torus converts any mean-curvature excess into a contradiction unless the claimed estimate holds.

What would settle it

Produce a fill-in of a flat metric $\gamma$ on $S^1 \times T^{n-2}$ with $R_g \ge -n(n-1)$ and $H_{\Sigma} > 0$ for which the normalized total mean curvature exceeds $\frac{1}{2}\left(\frac{4\pi}{n\sigma}\right)^n$; a numerical construction in the allowed dimensions would disprove the sharp estimate. Separately, check the rescaling step in the interpolation theorem: for any glued metric $g'$ with boundary $\gamma$, a global rescaling $c^2g'$ has boundary $c^2\gamma$, so preserving the boundary metric forces $c=1$; verifying whether a different deformation can raise the scalar-curvature lower bound to $-n(n-1)$ would decide whether the arbitrary-boundary theorem stands.

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Extended reading notes

Core claim

The central claim is Theorem 1.6: if $\gamma$ is flat on $\Sigma$, $g$ fills $\Omega = B^2 \times T^{n-2}$, $R_g \ge -n(n-1)$, $g|_{\partial\Omega} = \gamma$, and $H_{\Sigma} > 0$, then $\frac{1}{\mathrm{vol}(\Sigma)} \int_{\Sigma} (H_{\Sigma} - (n-1))\, d\mathrm{vol}_{\gamma} \le \frac{1}{2} \left(\frac{4\pi}{n\sigma}\right)^n$, where $\sigma$ is the length of the shortest closed curve in $\Sigma$ whose integral of $\Xi$ is nonzero, and $\Xi$ is the pullback of the volume form on the circle factor. The constant is sharp: taking larger and larger pieces of the model metrics on $\mathbb{R}^2 \times T^{n-2}$ makes the left-hand side approach the bound. The paper also proves an interpolation theorem comparing different boundary metrics and uses it to obtain a uniform total-mean-curvature bound for arbitrary boundary metrics on the same torus.

Load-bearing premise

The proof of the interpolation theorem ends by asserting that the glued metric can be rescaled so its scalar curvature is at least $-n(n-1)$, but a constant rescaling changes the boundary metric from $\gamma$ to $c^2\gamma$, so that step is not justified as written; if it cannot be repaired, the uniform bound for arbitrary boundary metrics does not follow.

Editorial extensions

If this is right

The sharp estimate settles the flat-torus case of the total-mean-curvature conjecture for fill-ins with scalar curvature bounded below.
For any boundary metric on $S^1 \times T^{n-2}$, the interpolation theorem yields a constant $C = C(n, \gamma)$ such that every admissible fill-in has $\int_{\partial\Omega} H\, d\mathrm{vol} \le C$.
The constant in the flat case is optimal, approached by the model metrics on $\mathbb{R}^2 \times T^{n-2}$ filling larger tori.
The proof demonstrates that the systolic inequality can replace the positive mass theorem in deriving total-mean-curvature estimates in this setting.

Reading between the lines

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

A direct proof via a positive mass theorem may be possible; the paper notes it used the systolic inequality instead, so recovering the same constant from a mass inequality would likely extend the result beyond the dimension range allowed by the gluing construction.
The interpolation mechanism suggests a route to sharp constants for non-flat boundary metrics: compare any boundary metric with a flat one, so the flat constant may control the whole conformal class up to a geometric factor.
The construction is testable numerically for small $n$: a sequence of fill-ins approaching the bound should develop a long cylindrical neck whose length is controlled by the systolic length $\sigma$ of the boundary torus.
The dimension restriction $3 \le n \le 7$ appears to come from the gluing lemma, so a different gluing argument is a natural route to higher-dimensional analogues.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper has two distinct layers. The core result, Theorem 1.6, is a genuinely new sharp total-mean-curvature estimate for flat torus boundaries in dimensions 3 through 7. The proof is a coherent combination of the Shi–Wang–Wei prescribed scalar curvature equation, a priori estimates, gluing, and the Brendle–Hung systolic inequality. The perturbation argument via the function f is clean, and the Horowitz–Myers example for sharpness is a good check. This layer is solid, and I would trust it after a routine fix of one typo: the final displayed inequality in the proof of Theorem 1.6 has the ratio inverted. Swapping (1−ε/4)/(1−ε/2) for (1−ε/2)/(1−ε/4) makes the contradiction go through, so this is repairable and not a real mathematical flaw.

The soft spot is exactly where the stress-test lands. In the proof of Theorem 1.7, after gluing the collar, the scalar curvature is only bounded below by min{−K, −n(n−1)} − 1. The last paragraph says: “Finally, we can rescale g so that the scalar curvature is bounded from below by −n(n−1).” A global rescaling multiplies the boundary metric by c². Since Theorem 1.7 requires the boundary to be exactly γ, this step is unjustified whenever the constant K is large. Such pairs (γ̂, γ) exist on tori without any special pathology. Consequently, Theorem 1.7 is not established, and Theorem 1.8, which depends on it, is unsupported. This is not a minor gap: the abstract advertises a general total-mean-curvature bound, and that claim rests on Theorem 1.8.

The reliance on the Brendle–Hung preprint [5] is real but not circular. It is an external dependency on an unpublished proof, which makes the result contingent, but the paper does not assume its own conclusion. The author should cite the published version when it appears.

Who is this for? People working on scalar curvature fill-ins and quasi-local mass. The flat-torus case is the clean result; the interpolation claim needs more work. I would send it to a serious referee because the core theorem is important and likely correct, and the gap is addressable. The author should fix the typo, repair or restrict the rescaling argument in Theorem 1.7, and resubmit.

Referee Report

2 major / 4 minor

Summary. The paper studies fill-ins of the torus Σ = S^1 × T^{n-2} (3 ≤ n ≤ 7) with scalar curvature bounded below by -n(n-1). Theorem 1.6 establishes, for flat boundary metrics γ, a sharp upper bound on the total mean curvature of the boundary, with the constant expressed in terms of the systolic length σ; sharpness is claimed via Horowitz–Myers metrics. Theorem 1.7 is an interpolation result: from a fill-in of a smaller metric γ̂ one constructs a fill-in of a larger metric γ with controlled total mean curvature. Theorem 1.8 then derives a general total-mean-curvature bound for arbitrary boundary metrics by combining Theorems 1.6 and 1.7. The proofs use the prescribed scalar curvature equation, a priori estimates, a gluing construction, and the Brendle–Hung systolic inequality.

Significance. If correct, Theorem 1.6 resolves a special case of Gromov's conjecture on total mean curvature for fill-ins with scalar curvature bounded below, in the setting of flat tori, and provides a sharp constant. The use of the Brendle–Hung systolic inequality in place of a direct positive mass argument is a genuine novelty. The paper also gives a route toward general boundary metrics via Theorem 1.8. The arguments are largely coherent, and the dependence on machine-checkable or previously established estimates (parabolic a priori estimates, gluing lemmas) is explicit. However, two issues affect the proofs as written: a typo in the final inequality of Theorem 1.6, and a more substantial gap in the rescaling step of Theorem 1.7.

major comments (2)

[Section 3, Proof of Theorem 1.7, last paragraph] The assertion 'Finally, we can rescale g so that the scalar curvature is bounded from below by -n(n-1)' is not justified. A global rescaling g ↦ c²g changes the boundary metric from γ to c²γ, so the required boundary condition g|_{∂Ω} = γ forces c = 1. When the constant K = sup_{Σ×[0,1]} |R_{γ_t}| > n(n-1), the glued metric satisfies only R_g ≥ -K - 1 < -n(n-1), and no constant rescaling can raise the scalar curvature lower bound to -n(n-1) while preserving the boundary metric exactly. Since Theorem 1.8 relies on Theorem 1.7, this is a load-bearing gap. The statement of Theorem 1.7 may still be true, but the proof as written does not establish it.
[Proof of Theorem 1.6, final displayed inequality] The displayed inequality 'hat σ λ^{-1} ≤ (1 - ε/4)/(1 - ε/2) σ' has the ratio inverted. Combining the lower bound on \tilde H - (n-1) with the systolic upper bound yields hat σ λ^{-1} ≤ (1 - ε/2)/(1 - ε/4) σ, not the displayed inequality. With the corrected ratio, choosing λ large gives hat σ/λ ≤ (1 - ε/2)/(1 - ε/4) σ < σ, which contradicts the convergence of hat σ/λ to σ as λ → ∞. As written, the displayed inequality does not yield a contradiction, so the proof is incomplete at this point; however, the error appears to be a localized algebra mistake that is repairable from the preceding inequalities.

minor comments (4)

[Proof of Theorem 1.8] The notation 'C = C(n,γ,γ′)' should be 'C = C(Σ,γ,\hat γ,n)' or similar; also the proof refers to 'γ′' which is not defined at that point.
[Lemma 3.1, proof] In the display of the comparison ODEs, there is an extra comma after 'w_-(0) = min Σ v0, ,' and the expression for w_+ should be 'w_+(0) = max_Σ v0'. These typos should be corrected for clarity.
[Proof of Theorem 1.6] Please clarify the precise form of the Brendle–Hung systolic inequality used. The proof applies it as a pointwise bound on \tilde H - (n-1). If the original statement is an integral inequality, the pointwise consequence should be justified using the near-constancy of the mean curvature on \hat Σ_λ established in Proposition 2.8.
[Throughout] There are numerous typographical issues, e.g., 'g|∂Ω =gRn|∂Ω' in Theorem 1.1, 'the g outward unit normal vector' in Theorem 1.6, and 'Letfε to be the unique solution' in the proof of Theorem 1.6. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main constant is derived from an external systolic inequality, and no fitted parameter or self-referential definition is used.

full rationale

The derivation chain for Theorem 1.6 is self-contained from the prescribed scalar curvature equation, the a priori estimates, the gluing lemma, and the external Brendle–Hung systolic inequality [5]. The constant (1/2)(4π/(nσ))^n is not fitted: it is the sharp constant appearing in that systolic inequality, and sharpness is independently checked against Horowitz–Myers metrics. The proofs of Theorems 1.7 and 1.8 are interpolation and reduction arguments based on the Shi–Wang–Wei collar construction and the same gluing lemma; they do not assume the total-mean-curvature bound being proved. The only flagged passage is the last sentence of the proof of Theorem 1.7 (Section 3): 'Finally, we can rescale g so that the scalar curvature is bounded from below by −n(n−1).' This is a genuine scaling/correctness gap, because a global rescaling changes the boundary metric from γ to c²γ, but it is not circularity: it does not make the theorem's conclusion an input of its own proof. No self-citation by the author is load-bearing: [5] is a preprint by Brendle and Hung, not by the author, and [21] is an external result. There is no renaming of a known result, no fitted input called a prediction, and no uniqueness theorem imported from the authors. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No numerical parameters are fitted; the only external inputs are the systolic inequality of Brendle-Hung and the PDE construction of Shi-Wang-Wei. The paper introduces no new particles, forces, or entities.

assumptions (3)

domain assumption Brendle-Hung systolic inequality for compact fill-ins of S^1 x T^(n-2) with scalar curvature >= -n(n-1) and boundary mean curvature >= n-1 (cited as [5]).
The proof of Theorem 1.6 invokes this inequality as a black box at the end of Section 2; if the inequality's hypotheses are not met or its statement differs, the contradiction argument fails.
domain assumption Shi-Wang-Wei prescribed-scalar-curvature interpolation construction for boundary metrics gamma_hat and gamma (Lemma 3.1, summarized from [21]).
Theorem 1.7 relies on Lemma 3.1, which is stated as a summary of [21]; the current paper sketches the proof but does not independently establish global existence and monotonicity beyond the cited reference.
standard math Standard parabolic regularity, Krylov-Safonov and Schauder estimates, and maximum principle arguments used in Lemmas 2.3 through 2.6.
These are standard tools invoked without proof to obtain a priori bounds and derivative estimates for the prescribed scalar curvature equation.

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Cite this review

Pith. "Pith review of Fill-Ins of Tori with Scalar Curvature Bounded from Below." pith.science (2026). https://pith.science/paper/7VO52P7P

@misc{pith2026241114667,
  author       = {Pith},
  title        = {Pith review of: Fill-Ins of Tori with Scalar Curvature Bounded from Below},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VO52P7P}},
  note         = {Machine review of arXiv:2411.14667}
}

abstract

Let $\gamma$ be a Riemannian metric on $\Sigma = S^1 \times T^{n-2}$, where $3 \leq n \leq 7$. Consider $\Omega = B^2 \times T^{n-2}$ with boundary $\partial \Omega = \Sigma$, and let $g$ be a Riemannian metric on $\Omega$ such that the scalar curvature $R_g \geq -n(n - 1)$ and $g|_{\partial \Omega} = \gamma$. Assuming the mean curvature of $\partial \Omega$ with respect to the outward normal is positive, we establish that the total mean curvature of $\partial \Omega$ is bounded from above by a constant depending only on $n$ and $\gamma$. Furthermore, we compute the sharp constant for this estimate when $\gamma$ is a flat metric. This result resolves a special case of a conjecture by Gromov concerning total mean curvature of fill-in with scalar curvature bounded from below. The proof combines techniques developed by Shi-Tam, Shi-Wang-Wei, as well as recent work by Brendle-Hung on the systolic inequality.

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Brendle, S., and Hung, P. K. Systolic inequalities and the horowitz-myers con- jecture, 2024

2024

[2] [1]

Andersson, L., Cai, M., and Gallow ay, G. J. Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincar´ e 9, 1 (2008), 1–33

work page 2008

[3] [2]

Boundary conditions for scalar curvature

B¨ar, C., and Hanke, B. Boundary conditions for scalar curvature. In Perspectives in scalar curvature. Vol. 2 . World Sci. Publ., Hackensack, NJ, [2023] ©2023, pp. 325– 377

work page 2023

[4] [3]

Quasi-spherical metrics and prescribed scalar curvature

Bartnik, R. Quasi-spherical metrics and prescribed scalar curvature. In Selected works of Robert A. Bartnik . Int. Press, Somerville, MA, [2021] ©2021, pp. [333–373]. Reprint of [1198599]

work page 2021

[5] [4]

Rigidity phenomena involving scalar curvature

Brendle, S. Rigidity phenomena involving scalar curvature. In Surveys in diﬀerential geometry. Vol. XVII , vol. 17 of Surv. Diﬀer. Geom. Int. Press, Boston, MA, 2012, pp. 179–202

work page 2012

[6] [6]

C., and Neves, A

Brendle, S., Marques, F. C., and Neves, A. Deformations of the hemisphere that increase scalar curvature. Invent. Math. 185 , 1 (2011), 175–197

work page 2011

[7] [7]

D., and York, Jr., J

Brown, J. D., and York, Jr., J. W. Quasilocal energy in general relativity. In Mathematical aspects of classical ﬁeld theory (Seattle, WA , 1991) , vol. 132 of Con- temp. Math. Amer. Math. Soc., Providence, RI, 1992, pp. 129–142

work page 1991

[8] [8]

D., and York, Jr., J

Brown, J. D., and York, Jr., J. W. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47 , 4 (1993), 1407–1419

work page 1993