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arxiv: 2407.21312 · v4 · pith:APCBZ5MMnew · submitted 2024-07-31 · 🧮 math.DG

Scalar curvature rigidity of spheres with subsets removed and L^infty metrics

Jinmin Wang , Zhizhang Xie This is my paper

Pith reviewed 2026-05-23 22:43 UTC · model grok-4.3

classification 🧮 math.DG
keywords scalar curvature rigidityL^∞ metricsspheres with subsets removedwrapping propertypositive mass theoremasymptotically flat manifolds
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The pith

Scalar curvature rigidity holds for L^∞ metrics on spheres minus closed subsets of codimension at least n/2+1 that satisfy the wrapping property.

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

The paper proves that any L^∞ metric of nonnegative scalar curvature on the n-sphere with a closed subset Σ removed must be the standard round metric, provided Σ has codimension at least n/2+1 and satisfies the wrapping property. This extends classical rigidity theorems to metrics with limited regularity and to domains with certain singularities removed. The wrapping property, which holds for subsets contained in a hemisphere or for finite sets, is the key condition that lets the argument go through. The same methods give an analogous rigidity statement on tori and, as a corollary, a positive mass theorem for complete asymptotically flat spin manifolds that may have arbitrarily many ends.

Core claim

For an L^∞ metric g on S^n ∖ Σ with nonnegative scalar curvature, where Σ is closed, has codimension at least n/2 + 1, and satisfies the wrapping property, g must be isometric to the standard round metric on the sphere.

What carries the argument

The wrapping property of the removed set Σ, a condition that permits the metric to be extended or compared across the singularity in a controlled way for the rigidity argument.

If this is right

  • Scalar curvature rigidity extends from smooth metrics to L^∞ metrics on the punctured sphere under the stated conditions on Σ.
  • An analogous rigidity result holds for L^∞ metrics on tori that are smooth away from subsets of codimension at least n/2 + 1.
  • A positive mass theorem holds for complete asymptotically flat spin manifolds with arbitrary numbers of ends when the metric is merely L^∞.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that rigidity phenomena may persist for metrics with singularities of even lower regularity if the wrapping property can be verified.
  • It opens the possibility of applying similar techniques to other curvature conditions or to manifolds with more general ends.

Load-bearing premise

The removed set Σ must satisfy the wrapping property and have codimension at least n/2 + 1.

What would settle it

An explicit L^∞ metric of nonnegative scalar curvature on S^3 minus a circle that is not isometric to the round metric would falsify the claim.

read the original abstract

We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of wrapping property was introduced by the second author for studying related scalar curvature rigidity problems on spheres. For example, any closed subset of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb S^n$ satisfy the wrapping property. The same techniques also apply to prove an analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth away from certain subsets of codimension at least $\frac{n}{2}+1$. As a corollary, we obtain a positive mass theorem for complete asymptotically flat spin manifolds with arbitrary ends for $L^\infty$ metrics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves scalar curvature rigidity for L^∞ metrics on S^n ∖ Σ (n ≥ 3), where Σ is a closed subset of codimension at least n/2 + 1 satisfying the wrapping property (introduced by the second author). Examples include subsets in a hemisphere or finite sets. The techniques extend to tori with similar removals, and a corollary yields a positive mass theorem for complete asymptotically flat spin manifolds with arbitrary ends under L^∞ metrics.

Significance. If the results hold, they extend scalar curvature rigidity theorems to low-regularity (L^∞) metrics away from controlled singular sets, building on prior work by the second author. The positive mass corollary for manifolds with arbitrary ends is a notable application. The manuscript ships no machine-checked proofs or code, but the codimension/wrapping hypotheses are explicitly stated and the approach appears independent of fitted parameters.

minor comments (3)
  1. §1 (Introduction): the statement of the wrapping property is referenced to prior work but a self-contained definition or key properties used in the proof would improve readability for readers unfamiliar with the second author's earlier papers.
  2. The positive mass corollary is stated in the abstract and introduction but the precise reduction from the sphere rigidity result to the asymptotically flat setting is not detailed in the provided sections; adding a short outline or reference to the relevant theorem would strengthen the exposition.
  3. Notation for the L^∞ metric and the definition of scalar curvature in the distributional sense should be clarified early, as the extension from smooth to L^∞ regularity is central.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no individual points requiring rebuttal or clarification at this stage. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes scalar curvature rigidity for L^∞ metrics on S^n minus a closed set Σ of codimension ≥ n/2+1 satisfying the wrapping property (introduced in prior work by the second author). The central derivation adapts existing techniques to the L^∞ setting and obtains a positive-mass corollary; the wrapping property functions as an explicit hypothesis rather than a derived or fitted quantity. No equation or step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem is imported solely via self-citation. The result remains independent of the present paper's fitted values and is externally falsifiable via the stated geometric assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; full list of background assumptions and any free parameters cannot be extracted. Standard differential geometry axioms are presumed.

axioms (1)
  • standard math Standard assumptions of differential geometry on manifolds, metrics, and scalar curvature.
    Implicit in any statement about scalar curvature rigidity on Riemannian manifolds.

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Reference graph

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