Positive scalar curvature and isolated conical singularity

Changliang Wang; Xianzhe Dai; Yukai Sun

REVIEW 2 major objections 5 minor 1 cited by

Positive scalar curvature and isolated conical singularity

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Pith's one-line read The paper proves that no conically singular metric on $M\#\mathbb{T}^n$ can have positive scalar curvature, and that nonnegative scalar curvature forces flatness plus smooth extension.

desk verdict Main non-existence theorem is solid and new; the arbitrary-cross-section rigidity claim rests on a false local simple-connectivity assumption and needs repair. read the letter →

arxiv 2412.02941 v1 pith:GRH5ZA52 submitted 2024-12-04 math.DG

Xianzhe Dai , Yukai Sun , Changliang Wang This is my paper · ORCID

classification math.DG MSC 53C2153C2453C2358J50

keywords scalarcurvatureisolatedconicalsingularityGerochconjecturetorusconformalblow-uprigidityRCDspacesasymptoticallyflatmanifolds

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 proves that the classical obstruction to positive scalar curvature on tori survives when the metric is allowed to have isolated conical singularities. On any connected sum $M\#\mathbb{T}^n$ with $n\ge 3$, under a mild topological hypothesis ($n\le 7$ or $M$ spin), there is no Riemannian metric whose regular part has nonnegative scalar curvature that is positive at some point. It also proves rigidity: if such a metric has merely nonnegative scalar curvature, it must be flat and smoothly extend across each singularity. This matters because it transfers a foundational smooth-manifold theorem to singular and incomplete geometries, where positive scalar curvature questions arise in the study of weak notions of curvature and in positive mass theorems for singular spaces.

What carries the argument

The mechanism is conformal blow-up. The paper finds a positive solution $u$ of the conformal Laplace equation $-\Delta_g u + \frac{n-2}{4(n-1)}\mathrm{Sc}_g u=0$ that grows like $r^{2-n}$ near a conical tip and has the standard Green's-function falloff $1+A\rho^{2-n}$ at an asymptotically flat end. The conformally changed metric $\tilde g = u^{4/(n-2)}g$ then satisfies $\mathrm{Sc}_{\tilde g}=u^{-4/(n-2)}\mathrm{Sc}_g$, so nonnegative scalar curvature is preserved, while the change of variable $s=1/r$ turns the singular tip into a complete end modeled on $ds^2+s^2g_N$. The resulting complete metric with positive scalar curvature contradicts the known torus obstruction for complete metrics, and this contradiction is the engine of both non-existence proofs. For the rigidity half, the engine is different: deform a scalar-flat metric in the direction of its Ricci tensor, use the positivity of the first eigenvalue of the conformal Laplacian on conical manifolds to produce a positive-scalar-curvature deformation, and then invoke RCD-space fundamental-group rigidity to conclude the metric is a flat torus.

What would settle it

A concrete counterexample would settle the matter: a metric on $\mathbb{T}^3$ with a single isolated conical singularity and positive scalar curvature on the regular part, or on $M\#\mathbb{T}^n$ in any allowed dimension. A more targeted check is to solve for the harmonic function used in the blow-up on a cone with a non-spherical cross section and test numerically whether the required asymptotics hold.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a closed smooth $n$-manifold $M$, $n\ge 3$, with either $n\le 7$ or $M$ spin, the manifold $M\#\mathbb{T}^n$ admits no Riemannian metric $g$ with finitely many isolated conical singularities such that $\mathrm{Sc}_g\ge 0$ on the regular part and $\mathrm{Sc}_g>0$ at some point. Here 'positive scalar curvature' is used in that nonnegative-and-somewhere-positive sense. The same theorem asserts rigidity: a conically singular metric on $M\#\mathbb{T}^n$ with nonnegative scalar curvature must be scalar flat, then Ricci flat, then flat, and must extend smoothly across the singular points; in particular the cross sections of the singularities are forced to be round spheres even when they were not assumed spherical. A parallel statement treats compact manifolds with boundary whose boundary components are pinched to conical points, and the torus can be replaced by any $\Lambda^2$-enlargeable closed manifold in the spin case.

Load-bearing premise

The load-bearing premise is that on every asymptotically flat manifold with a conical singularity of the type considered, there exists a harmonic function with exactly the growth $r^{2-n}$ near the cone tip and $1+A\rho^{2-n}$ at infinity; if that existence statement fails for some allowed cross section, the first non-existence proof collapses.

Editorial extensions

If this is right

For every closed manifold $M$ with $3\le n\le 7$, or spin $M$ of any dimension, $M\#\mathbb{T}^n$ carries no conically singular metric with positive scalar curvature on the regular part.
Tori and, more generally, $\Lambda^2$-enlargeable closed manifolds admit no metric with finitely many isolated conical singularities and positive scalar curvature.
Any conically singular metric on $M\#\mathbb{T}^n$ with nonnegative scalar curvature is flat and extends smoothly through every singular point, with spherical cross sections forced by the rigidity conclusion.
For compact manifolds with boundary, pinching each boundary component to a conical point still forbids positive scalar curvature, and nonnegative scalar curvature forces flatness and spherical boundary components.
The compactification argument used to derive positive mass theorems from torus obstructions carries over to asymptotically flat manifolds with isolated conical singularities.

Reading between the lines

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

The conformal blow-up strategy suggests an extension to singular sets of codimension at least 3: if a Green's-type function with the right asymptotics exists, the same contradiction should rule out positive scalar curvature for those singularities.
The rigidity argument could plausibly generalize from isolated points to conical strata of codimension at least 3; the missing ingredient would be the appropriate eigenfunction asymptotics and RCD structure for stratified spaces.
One can test sharpness by placing conical singularities on manifolds with positive Yamabe invariant rather than on $M\#\mathbb{T}^n$; the theorem says nothing there, and explicit examples would show that topology, not just singularity type, drives the obstruction.

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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

Three things you should know. First, the main non-existence theorem is real: for n ≤ 7 or spin, M#T^n admits no metric with isolated conical singularities and positive scalar curvature on the regular part. The proof via conformal blow-up is standard in outline but the analytic setup on conifolds — harmonic functions with the right asymptotics near the singularity and at infinity — is handled carefully. The alternative proof through the conformal Laplacian on the compact conifold is a nice addition, and the Schoen-Yau-Lohkamp compactification appendix is a useful contribution. The rigidity statement for spherical cross-sections (scalar-flat implies flat and smoothly extendable) is also new, and I see no gap there.

Second, the advertised generalization to non-spherical links has a genuine gap. In Section 5, after proving Ric = 0, the paper applies Mondino-Wei's RCD rigidity and needs the metric space to be locally simply connected. The text says this is "clear." It is not clear; it is false for non-spherical links. For a link like RP^3 in dimension 4, small punctured neighborhoods have the homotopy type of RP^3, so the space is not locally simply connected. The usual fundamental group does not control Mondino-Wei's revised fundamental group, and the paper gives no argument that the T^n generators survive in that revised group. This does not affect the main Theorem 1.1 in the manifold-point case (where the link is S^{n-1}, simply connected for n ≥ 3), nor the non-existence part for arbitrary links. But Theorem 1.5 and the abstract's claim about arbitrary cross-sections rely on this step. The authors need to either supply the missing argument or weaken the rigidity claim to links that are simply connected.

Third, the citation pattern looks fine. The paper relies on the authors' earlier analytic framework, but those results are independent and published separately; no circularity.

Who is this for? Geometric analysts working on scalar curvature and singular spaces. The main theorem is a substantial extension of Geroch's conjecture and deserves a serious referee. I would send it out, with a clear request to fix or trim the arbitrary-link rigidity claim.

Referee Report

2 major / 5 minor

Summary. The paper proves Geroch-type theorems for Riemannian metrics with isolated conical singularities on manifolds of the form M#T^n. The main non-existence claim (Theorem 1.1) is that for n>=3, with either n<=7 or M spin, there is no metric on M#T^n with finitely many isolated conical singularities whose scalar curvature on the regular part is nonnegative and strictly positive somewhere. The rigidity claim is that if such a metric has nonnegative scalar curvature and is scalar flat, then it is flat and extends smoothly across the singularities. The authors give two proofs of non-existence, both by conformally blowing up the singular point(s) to obtain a complete metric and then invoking known smooth theorems (Chodosh-Li, Wang-Zhang). The rigidity proof first shows Ricci flatness via a deformation argument using the conformal Laplacian, then applies the RCD rigidity theorem of Mondino-Wei. Theorem 1.5 extends the statement to compact manifolds with boundary whose boundary components are pinched to conical points, allowing non-spherical links.

Significance. If the arguments are completed, the paper would be a valuable extension of the Geroch conjecture and of Li-Mantoulidis-type rigidity to isolated conical singularities, including topological singularities with non-spherical links. The two independent conformal blow-up proofs are clearly structured, and the reduction of the rigidity problem to RCD rigidity is elegant. The paper also draws on a substantial analytic framework from the authors' prior work on conifolds, which is published separately. The main value lies in the combination of conformal analysis on conifolds with known smooth scalar-curvature obstructions and synthetic Ricci curvature rigidity. However, two load-bearing gaps currently prevent the stated theorems from following from the proofs as written.

major comments (2)

[§2 and §4 (proofs of Theorem 1.1)] The proof of the non-existence part of Theorem 1.1 is written for a single conical singularity: Lemma 2.4 and Lemma 4.1 are stated for one cone point, and the proofs begin with a manifold (M#T^n, g, o) with a single singularity. Theorem 1.1 and Corollary 1.3 claim the result for finitely many isolated conical singularities, but no reduction to the single-singularity case is supplied. If there are k>1 singularities and the conformal factor has a pole at only one of them, the conformally changed metric is still genuinely singular at the other k-1 points, so the resulting object is not a smooth complete manifold and the cited theorems in [5] and [39] do not apply. The authors should either prove a multi-pole version of the harmonic function or Green's function with prescribed leading asymptotics at every cone tip, or restrict the theorem to a single isolated singularity.
[§2, Eq. (2.9) and §4, Eq. (4.10)] The paper defines 'positive scalar curvature' as Sc>=0 and strictly positive somewhere, and the proofs of non-existence produce a complete metric with nonnegative scalar curvature that is strictly positive at some point. The contradiction is then invoked with Theorem 3 of [5] and Theorem 1.1 of [39], which in the literature are standardly stated for pointwise positive scalar curvature (Sc>0 everywhere). Since the constructed scalar curvature can vanish on large sets, the cited theorems do not directly rule out the constructed metric. The authors need either to cite versions of [5,39] that apply to the weaker condition, or to add a conformal deformation argument that promotes the constructed metric to one with pointwise positive scalar curvature while preserving completeness. This is a load-bearing step in both non-existence proofs and hence also in the rigidity conclusion.

minor comments (5)

[§5] The assertion that the metric space is locally simply connected is correct even for non-spherical links: small balls about the cone tip are truncated cones, which are contractible, and small balls about regular points are Euclidean coordinate balls. The authors may wish to add one sentence making this explicit, since the point is currently asserted rather than justified.
[Abstract and Theorem 1.1] The abstract's first sentence says 'an isolated conical singularity', while Theorem 1.1 says 'finitely many isolated conical singularities'; please align the terminology, especially in light of the first major comment.
[§2, Lemma 2.4] Lemma 2.4 quotes Lemma 4.1 of [10] without restating its hypotheses. Since the entire first proof depends on this lemma, the authors should spell out the geometric and analytic conditions on the asymptotically flat manifold (e.g., nonnegative scalar curvature, dimension, asymptotic order) so that the reader can verify that the surgically constructed manifold satisfies them.
[§2, Definition 2.1] Definition 2.1 defines only a single conical singularity; the finitely many case is merely said to be analogous. For a rigorous statement of Theorem 1.1, the definition should be extended explicitly to finitely many points, including the decay condition for the perturbation h near each point.
[Appendix A, Proposition A.1] There is a typo 'Green's funcction' in Proposition A.1, and some reference titles contain typos (e.g., 'Compacitiﬁcation' in [8]). Please proofread the bibliography.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the nonexistence proof reduces to external PSC obstructions, and the self-citations are technical analytic lemmas rather than restatements of the target result.

full rationale

The central nonexistence claim (Theorem 1.1) is proved by conformally blowing up a hypothetical conically singular PSC metric on M#T^n to a complete PSC metric and then invoking external obstructions: Chodosh-Li [5] for n <= 7 and Wang-Zhang [39] for spin M. In the first proof, the conformal factor is supplied by Lemma 2.4, quoted from the authors' prior paper [10]. This is a load-bearing self-citation, but the lemma is a parameter-free existence statement for a harmonic function with prescribed asymptotics on an asymptotically flat conifold; it does not assume nonexistence of PSC metrics or any equivalent of Theorem 1.1. The second proof develops its own weighted Sobolev theory on compact conifolds (Propositions 3.1-3.6), borrowing Fredholm and surjectivity arguments from [10] only as analytic scaffolding. The rigidity argument legitimately uses the already-proved nonexistence part to reduce to scalar flatness, then deforms by the conformal Laplacian; the spectral tools cited from [11] and [13] are prior analytic results on conifolds, not restatements of the target conclusion, and the final flat-torus step invokes external RCD rigidity from Bertrand-Ketterer-Mondello-Richard [1] and Mondino-Wei [30]. No fitted parameter is renamed as a prediction, and no definition is made in terms of the target theorem. I therefore find no circular step. Separately, as a correctness concern rather than circularity, Section 5 asserts that the metric completion 'clearly is a connected and locally simply connected' space; for a non-spherical link N with nontrivial pi_1(N), arbitrarily close to the singularity there are neighborhoods homeomorphic to N x (0, epsilon), so this assertion is false. That threatens the rigidity claim in Theorem 1.5 for arbitrary cross-sections, but it does not affect the nonexistence part or the spherical-link case of Theorem 1.1.

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

No free parameters or invented entities. The paper rests on a chain of established but nontrivial results, several from the authors' own prior work, which serve as the analytic foundation.

assumptions (5)

domain assumption Existence of harmonic functions with prescribed asymptotics on AF manifolds with isolated conical singularities (Lemma 2.4, [10]).
This prior result by the same authors is used in the first proof of non-existence to construct the conformal factor.
domain assumption Spectral properties of the conformal Laplacian on compact manifolds with isolated conical singularities ([11], [13]).
Used in the rigidity proof to deform along Ricci and obtain a PSC metric.
domain assumption Theorem A in [1]: conical Riemannian manifolds with Ricci curvature bounded below are RCD(0,n) spaces.
Used to enter the synthetic curvature framework for the rigidity conclusion.
domain assumption Corollary 1.4 in [30] (Mondino-Wei): RCD(0,n) spaces with n independent infinite-order fundamental group generators are flat tori.
Used to conclude the singular metric is flat and extends smoothly.
domain assumption No complete PSC metric on M#T^n: Chodosh-Li [5] for n <= 7, Wang-Zhang [39] for spin complete manifolds.
These external results provide the contradiction after conformal blow-up.

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Pith. "Pith review of Positive scalar curvature and isolated conical singularity." pith.science (2026). https://pith.science/paper/GRH5ZA52

@misc{pith2026241202941,
  author       = {Pith},
  title        = {Pith review of: Positive scalar curvature and isolated conical singularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRH5ZA52}},
  note         = {Machine review of arXiv:2412.02941}
}

abstract

We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical.

Figures

Figures reproduced from arXiv: 2412.02941 by the authors.

**Figure 1.** Conically pinching the boundary An important application of the resolution of the Geroch conjecture and its generalizations is the positive mass theorem. For smooth asymptotically flat manifolds, a well-known Schoen-Yau-Lohkamp compactification argument reduces the positive mass theorem to the Geroch conjecture for M#T n , where M is a compact smooth manifold. Kazaras [20] and Cecchini-Frenck-Zeidler [2] obtained t… view at source ↗

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[1] [5]

Generalized soap bubbles and the topolo gy of manifolds with positive scalar curvature

Otis Chodosh and Chao Li. Generalized soap bubbles and the topolo gy of manifolds with positive scalar curvature. Ann. of Math. (2) , 199(2):707–740, 2024

work page 2024

[2] [39]

On the generalized Geroch c onjecture for complete spin manifolds

Xiangsheng Wang and Weiping Zhang. On the generalized Geroch c onjecture for complete spin manifolds. Chin. Ann. Math. Ser. B , 43(6):1143–1146, 2022

work page 2022

[3] [1]

Stratiﬁed spaces and synthetic Ricci curvature bounds

J´ erˆ ome Bertrand, Christian Ketterer, Ilaria Mondello, and Th omas Richard. Stratiﬁed spaces and synthetic Ricci curvature bounds. Ann. Inst. Fourier (Grenoble) , 71(1):123–173, 2021

work page 2021

[4] [2]

Positive scalar curvature with point singularities

Simone Cecchini, Georg Frenck, and Rudolf Zeidler. Positive scalar curvature with point singularities. arXiv e-prints , page arXiv:2407.20163, July 2024

arXiv 2024

[5] [3]

Incompressible h ypersurface, positive scalar curvature and positive mass theorem

Jie Chen, Peng Liu, Yuguang Shi, and Jintian Zhu. Incompressible h ypersurface, positive scalar curvature and positive mass theorem. arXiv:2112.14442v1, 2021

arXiv 2021

[6] [4]

Positive scalar urvatu re metric and aspherical sum- mands

Shuli Chen, Jianchun Chu, and Jintian Zhu. Positive scalar urvatu re metric and aspherical sum- mands. arXiv:2312.04698v1, 2023

arXiv 2023

[7] [6]

The Dirac operator on spaces with conica l singularities and positive scalar curvatures

Arthur Weichung Chou. The Dirac operator on spaces with conica l singularities and positive scalar curvatures. Trans. Amer. Math. Soc. , 289(1):1–40, 1985

work page 1985

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Mass under the Ricci ﬂow

Xianzhe Dai and Li Ma. Mass under the Ricci ﬂow. Comm. Math. Phys. , 274(1):65–80, 2007

work page 2007