Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class

Jingbo Wan

REVIEW 2 major objections 4 minor 22 references

Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that every smooth metric with scalar curvature at least $\kappa$ is a local Burnett-class limit of smooth metrics with scalar curvature exactly $\kappa$; on closed manifolds this holds for every real $\kappa$, and on…

desk verdict A strong, likely correct proof of the Riemannian reverse-Burnett conjecture that deserves refereeing, with the main caveat being a disclosed but load-bearing dependence on an unpublished preprint. read the letter →

arxiv 2608.08707 v1 pith:IV26NREM submitted 2026-08-09 math.DG math.AP

Jingbo Wan This is my paper · ORCID

classification math.DGmath.AP MSC 53C2135J6058J05

keywords scalarcurvatureBurnettcompactnessconformalLaplacianreverseconjectureprescribedconvexintegrationW^{1∞}convergenceC^0stability

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 flexibility theorem for scalar curvature in a natural compactness regime for Riemannian metrics: sequences converge locally uniformly in $C^0$ while their first derivatives stay locally uniformly bounded, so the derivatives only converge in a weak sense and quadratic expressions can relax. In that class, every smooth metric with scalar curvature at least $\kappa$ is the limit of smooth metrics with scalar curvature exactly $\kappa$, whenever the manifold is closed or $\kappa\le 0$; for $\kappa<0$ and a complete target metric, the approximating metrics can be chosen complete. Combined with the $C^0$-stability theorem for scalar-curvature lower bounds, this gives the equality of the $C^{0,\alpha}_{\mathrm{loc}}$-closure of the exactly-$\kappa$ set with the at-least-$\kappa$ set for every $\alpha\in(0,1)$, and the paper proves that $\alpha<1$ is sharp. At $\kappa=0$, this is the Riemannian reverse-Burnett conjecture, strengthened by the uniform local $W^{1,\infty}$ bound and by the sharp Hölder exponent.

What carries the argument

Two mechanisms carry the argument. The first is a parameterized almost-prescription of scalar curvature: Proposition 3.3 produces a smooth one-parameter family $g_s$ starting at a given metric with $\mathrm{Scal}_{g_s}\approx \mathrm{Scal}_g-s^2k$, while staying arbitrarily close in $C^0$ and locally bounded in $W^{1,\infty}$; it is built from a two-scale corrugation in which one oscillation scale creates a signed quadratic term and a second scale cancels the unwanted affine term. The second is a global conformal correction via the conformal Laplacian $L_h=-c_n\Delta_h+\mathrm{Scal}_h$ with $c_n=4(n-1)/(n-2)$, using the conformal formula $\mathrm{Scal}_{u^{4/(n-2)}h}=u^{-(n+2)/(n-2)}L_h u$. Existence of the positive factor $u$ is obtained case-by-case: by continuity of the first eigenvalue of the conformal Laplacian, by parabolicity with a weighted spectral gap, by constant sub- and supersolutions, or by rescaling away a resonance and running a contraction argument.

What would settle it

Check Lemma 3.2 on a flat three-torus with a constant positive function $a$: compute the scalar curvature of the two-scale perturbation $h_F$ for increasing frequency $N$ and verify that $\|\mathrm{Scal}_{h_F}-\mathrm{Scal}_h+a^2\|_{C^0}$ tends to zero while $\|h_F\|_{W^{1,\infty}}$ stays bounded independently of $N$; an uncancelled $O(1)$ term or a derivative blow-up would falsify the local almost-prescription and with it Theorem 1.2.

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Extended reading notes

Core claim

The central claim is that the scalar-curvature inequality $\mathrm{Scal}_g\ge\kappa$ is exactly the relaxation of the equation $\mathrm{Scal}_g=\kappa$ under local Riemannian Burnett compactness. For a connected $n$-manifold with $n\ge 3$, if $M$ is closed or $\kappa\le 0$, every $g_0$ with $\mathrm{Scal}_{g_0}\ge\kappa$ is the local uniform limit of smooth metrics $\hat g_i$ satisfying $\mathrm{Scal}_{\hat g_i}=\kappa$ and locally uniformly bounded in $W^{1,\infty}$. The paper shows that the failure of the first derivatives to converge strongly carries the entire scalar-curvature defect: the divergence part of the curvature-density formula passes to the limit, while the quadratic part is what produces the inequality. It also proves sharpness: if a sequence of exactly-$\kappa$ metrics converges in $C^0$ and its derivatives converge strongly in $L^2_{\mathrm{loc}}$, the limit again has scalar curvature exactly $\kappa$, so the Hölder exponent $\alpha<1$ in the closure identity cannot be improved to Lipschitz.

Load-bearing premise

The load-bearing premise is that the local two-scale corrugation construction taken from reference [1] and used without proof is correct; if that construction cannot actually lower scalar curvature by approximately $s^2k$ while keeping the metrics $C^0$-close and first-derivative-bounded, then Proposition 3.3, and with it the whole theorem, would not follow.

Editorial extensions

If this is right

The closure identity $\overline{\{g:\mathrm{Scal}_g=\kappa\}}^{C^{0,\alpha}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\ge\kappa\}$ holds for every $\alpha\in(0,1)$, and the same equality holds in the $C^0_{\mathrm{loc}}$-topology; on closed manifolds the topologies may be taken globally.
At $\kappa=0$, every smooth metric of nonnegative scalar curvature on a connected manifold of dimension $n\ge 3$ is a local Riemannian Burnett limit of scalar-flat metrics, settling the reverse-Burnett conjecture in strengthened form.
For $\kappa<0$, the approximating metrics can be chosen complete whenever the original metric is complete.
The sharpness statement gives $\overline{\{g:\mathrm{Scal}_g=\kappa\}}^{C^{0,1}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g=\kappa\}$, so strong $C^{0,1}$ limits do not relax the equation.
For $\kappa>0$, the theorem covers closed manifolds, while the case of open manifolds is explicitly left open because the global conformal correction is not available.

Reading between the lines

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

Because the equality is sharp at the Hölder scale, intermediate closures should be visible if the compactness assumption is interpolated between $W^{1,\infty}$ and strong $W^{1,2}$; one could test whether $W^{1,p}$-bounded sequences with large $p$ produce a family of strict intermediate sets between $\{\mathrm{Scal}=\kappa\}$ and $\{\mathrm{Scal}\ge\kappa\}$.
The construction for $\kappa=0$ on open manifolds parabolicizes the metric and does not preserve completeness; a natural extension is to ask whether complete scalar-flat approximants exist for every complete metric of nonnegative scalar curvature.
The same two-scale cancellation is likely to apply to other equations whose curvature density splits into a divergence term plus a quadratic term in first derivatives, such as the time-symmetric constraints, which would connect this Riemannian flexibility statement to the initial-data side of the reverse Burnett problem.
For $\kappa>0$, the nonresonant rescaling shows that the obstruction on open manifolds is spectral rather than local; an extension would be to seek positive-$\kappa$ examples on open manifolds where the essential spectrum of the conformal Laplacian is pushed away from $\kappa/(n-1)$.

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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

Read Wan's paper. The headline result is real: for closed manifolds, or any manifold with κ≤0, every smooth metric with Scal≥κ is a limit in the local Riemannian Burnett class of metrics with Scal=κ, with uniform W^{1,∞} bounds. The corollary—the C^{0,α} closure of M_κ equals M_{≥κ} for every α<1—is sharp and genuinely surprising. At κ=0 this settles the Riemannian reverse-Burnett conjecture of Huneau and Luk. The proof splits into a local almost-prescription step (Section 3) and a global conformal correction; both are executed carefully. I checked the conformal corrections for κ<0 and the spectral argument for κ>0, and they look sound. The local step is load-bearing. Lemma 3.2 computes the two-scale cancellation explicitly, and the O(N^{-1}) bound follows from the definitions. But the final cancellation identity is exactly [1, Prop 2.2], an unpublished arXiv preprint. The paper states what it imports, which is honest, but a gap there would break every case of Theorem 1.2. My own reading of the computation suggests it is self-contained enough that the risk is low, yet I would want [1] checked or the key identity proved in an appendix before calling this definitive. The soft spots elsewhere are minor: Lemma 4.4's diagonal subsequence and Harnack propagation are sketched, and Lemma 4.1's uniform normalization is terse. These are standard and I did not find a real gap. The κ>0 open case is clearly flagged. Who should read this: anyone working on scalar curvature stability, Gromov's C^0 program, or high-frequency limits in GR. It deserves a serious referee: the main theorem is important, the proof is detailed, and the external dependence, while real, is disclosed and localized. I would accept it for review and, conditional on verifying [1], likely endorse publication.

Referee Report

2 major / 4 minor

Summary. Let M be a connected smooth n-manifold, n≥3. The paper proves (Theorem 1.2) that, if M is closed or κ≤0, every smooth metric g0 with Scal_{g0}≥κ is a limit, in the local Riemannian Burnett compactness class (C^0_loc convergence plus locally uniform W^{1,∞} bounds), of smooth metrics with constant scalar curvature κ. It then derives the C^{0,α}_loc closure identity {Scal_g=κ}^{C^{0,α}_loc}={Scal_g≥κ} for α<1, notes that α=1 is sharp, and at κ=0 states that this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk. The proof is organized as a parameterized almost-prescription theorem (Proposition 3.3), built on the two-scale convex-integration construction of [1], followed by global conformal corrections: eigenvalue and parabolicity arguments for κ=0, constant barriers for κ<0, and a nonresonant rescaling plus contraction argument for closed manifolds with κ>0.

Significance. If correct, this is a substantial and surprising flexibility result: it identifies the full scalar-curvature relaxation allowed by Gromov's C^0-stability theorem with the closure in the first-order Burnett compactness class, and it settles the Riemannian reverse-Burnett conjecture at κ=0 in a strengthened form. The paper is honest and careful in several places: the local scalar-curvature identity is computed explicitly, the uniform W^{1,∞} bounds are tracked, the open case κ>0 is openly left unresolved, and the sharpness at α=1 is proved. The main risk is structural rather than internal: the parameterized almost-prescription in Section 3 is explicitly built on the two-scale corrugation of the unpublished preprint [1], and the final cancellation identity in Lemma 3.2 is exactly the content of [1, Proposition 2.2]. If that external result contained a gap, Proposition 3.3 and all later sections would be unsupported.

major comments (2)

[§3, Lemma 3.2 and the preceding paragraph] The construction is introduced as 'based on the two-scale corrugation in [1, Proposition 2.2]', and the cancellation identity at the end of the proof of Lemma 3.2 is exactly the content of that proposition. Since [1] is an unpublished preprint, the correctness of Theorem 1.2 is structurally dependent on an external result that is neither stated as a separate lemma nor proved independently. The manuscript's own computation with (13) appears self-consistent, but it does not say which identities are imported from [1, Prop. 2.2] or verify them here. A sign error or a missed mean-zero condition in the imported two-scale cancellation would destroy the O(N^{-1}) estimate, and then the inductive error estimates (19)-(20) in Proposition 3.3, and hence every case of Theorem 1.2, would fail. I recommend proving the two-scale cancellation independently as a lemma, or at minimum stating the imported identities precisely and verifying them in this paper.
[§4.2, Lemma 4.4] The construction of the limit u_i for fixed i is compressed: the diagonal subsequence is asserted, and the uniformity of the Harnack constants as the exhaustion parameter j varies is not explicitly justified. The later claim that every subsequential limit is 1 also relies on positivity and local H^1 boundedness of u_i, which are established only after the Harnack propagation step. These steps are standard, but they are load-bearing for the open case κ=0, so the argument should be expanded into a fully explicit proof.

minor comments (4)

[§3, Lemma 3.2] The proof asserts that α_s has a uniform positive lower bound on the spatial support of a; this should be justified explicitly from compactness of supp_x a and uniform equivalence of the family (h_s).
[§4.2, proof of Theorem 1.2 for open M, κ=0] The sentence that the decrements Scal_{q_i}+m^{-1}β_i are bounded in C^0_loc is literally true only on K_{i+1}; the finitely many small i for which a given compact set is not contained in K_{i+1} are harmless, but the text should say this so that the uniform assertion in Proposition 3.3 is applied correctly.
[§4.1, closed case κ=0] The proof uses continuity of the first eigenvalue λ_1(g_{i,s}) in s to locate the root s_i; this is standard, but a one-line justification via min-max and the C^0 closeness (24) would make the argument self-contained.
[§6, Lemma 6.1] The convergence λ_ℓ(-Δ_{h_i})→λ_ℓ(-Δ_g) and the uniform lower bound λ_ℓ(-Δ_{h_i})≥c λ_ℓ(-Δ_g) are used to obtain the spectral gap; they follow from min-max and uniform equivalence of metrics, but stating the comparison lemma explicitly would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation uses external results forward, and the two-scale cancellation is recomputed locally.

full rationale

No circular reasoning detected. The central derivation chain is: Lemma 3.1 computes a localized scalar-curvature identity; Lemma 3.2 constructs the two-scale loops and verifies the cancellation with its own displayed algebra (the definition of sigma in (13) is chosen to make the final displayed error O(N^{-1}), not imported from the theorem being proved); Proposition 3.3 iterates Lemma 3.2 on a locally finite cover; Sections 4-6 combine Proposition 3.3 with independent conformal-correction arguments (eigenvalue crossing, parabolicity and weighted spectral gap, sub- and supersolutions, and a contraction mapping). Every load-bearing external input is used in the forward direction: [1] for the two-scale corrugation and almost-prescription framework, [9] and [2] for Gromov's C0-stability, [18] for the prior unparameterized almost-prescription result, [20] for the subcritical alternative, and [6] for elliptic estimates. None of these is authored by the present paper's author, and none assumes Theorem 1.2 or its corollaries. The dependence on the unpublished [1, Proposition 2.2] is explicitly disclosed and is a verification/correctness risk rather than a logical circularity; the manuscript's Lemma 3.2 provides a self-contained local computation, and no quoted equation reduces to the conclusion by construction. Accordingly the circularity score is 0.

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

The proof has no fitted parameters and introduces no new entities. It relies on standard elliptic PDE theory and two external theorems: the almost-prescription result of [1] and Gromov's C^0-stability theorem. The dependence on [1] is significant because that reference is a recent preprint, but it is disclosed.

assumptions (4)

standard math Mixed convex integration almost-prescription of Aliouane, Rifford, and Theillière [1, Theorem 1.1 and Proposition 2.2] is correct.
The local two-scale cancellation in Lemma 3.2 is explicitly based on [1, Proposition 2.2] and inherits its correctness; the paper does not reprove the full proposition.
standard math Gromov's C^0-stability theorem [9] (or Bamler's Ricci-flow proof [2]) gives the forward inclusion of the closure.
Used in Corollary 1.3 to show that the C^0 closure of Mκ is contained in M_{≥κ}; not needed for the main approximation theorem.
standard math Standard elliptic regularity, maximum principle, Harnack, and interior estimates of Gilbarg and Trudinger [6].
Used throughout Sections 2, 4, 5, and 6 for convergence of conformal factors and eigenfunctions.
domain assumption Manifold M is connected, smooth, without boundary, n≥3; if M is open then κ≤0; if M is closed, κ can be any real number.
These are the theorem's hypotheses in Theorem 1.2; the positive κ on open manifolds case is openly left unresolved.

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Cite this review

Pith. "Pith review of Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class." pith.science (2026). https://pith.science/paper/IV26NREM

@misc{pith2026260808707,
  author       = {Pith},
  title        = {Pith review of: Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IV26NREM}},
  note         = {Machine review of arXiv:2608.08707}
}

abstract

Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\kappa$ that are locally uniformly bounded in $W^{1,\infty}$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=\kappa\}}^{\,C^{0,\alpha}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geq\kappa\}, \quad \forall \alpha \in(0,1). \] The restriction $\alpha<1$ is sharp. At $\kappa=0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Almost prescribing scalar curvature by mixed convex integration

F. Aliouane, L. Rifford, and M. Theilli` ere,Almost prescribing scalar curvature by mixed convex integration, arXiv:2505.08384, 2025

work page Pith review arXiv 2025

[2] [2]

R. H. Bamler,A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature, Math. Res. Lett.23(2016), no. 2, 325–337

work page 2016

[3] [3]

G. A. Burnett,The high-frequency limit in general relativity, J. Math. Phys.30(1989), no. 1, 90–96

work page 1989

[4] [4]

De Lellis and L

C. De Lellis and L. Sz´ ekelyhidi, Jr.,High dimensionality andh-principle in PDE, Bull. Amer. Math. Soc.54(2017), no. 2, 247–282

work page 2017

[5] [5]

A. E. Fischer and J. E. Marsden,Deformations of the scalar curvature, Duke Math. J. 42(1975), no. 3, 519–547

work page 1975

[6] [6]

Gilbarg and N

D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, 2nd ed., Classics in Mathematics, Springer-Verlag, Berlin, 2001

work page 2001

[7] [7]

Guerra and R

A. Guerra and R. Teixeira da Costa,Oscillations in wave map systems and homoge- nization of the Einstein equations in symmetry, Arch. Ration. Mech. Anal.249(2025), Paper No. 9

work page 2025

[8] [8]

Gromov,Partial Differential Relations, Ergebnisse der Mathematik und ihrer Gren- zgebiete (3), vol

M. Gromov,Partial Differential Relations, Ergebnisse der Mathematik und ihrer Gren- zgebiete (3), vol. 9, Springer-Verlag, Berlin, 1986

work page 1986