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Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds

T0 review · 0 major / 3 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Topological linking at infinity forces polynomial decay of scalar curvature on non-compact manifolds of weakly bounded geometry.

desk verdict Linking at infinity forces polynomial scalar curvature decay on weakly bounded non-compact manifolds and yields end-localized obstructions via mu-bubbles and index theory. read the letter →

arxiv 2604.06547 v1 submitted 2026-04-08 math.DG math.APmath.GT

Shunichiro Orikasa This is my paper · ORCID
classification math.DGmath.APmath.GT
keywords non-compactmanifoldsscalarcurvaturepositivelinkingatinfinitydecaymu-bubblesminimalhypersurfacesindextheory
checked against alexander_duality_circle_linkingFoundation.ArithmeticFromLogic
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 shows that topological linking at infinity on complete non-compact manifolds with positive scalar curvature and weakly bounded geometry requires the scalar curvature to decay polynomially. This offers a topological explanation for the existence of metrics with quadratic scalar curvature decay. It further develops an obstruction theory for uniformly positive scalar curvature on specific ends of these manifolds by combining mu-bubble exhaustions with the study of stable minimal hypersurfaces and index theory. This matters because it links asymptotic topology to geometric constraints that may prevent certain manifolds from admitting metrics with scalar curvature bounded below by a positive constant.

What carries the argument

Topological linking at infinity, a condition on the ends that constrains asymptotic geometry and induces the required curvature decay.

What would settle it

A complete non-compact manifold of weakly bounded geometry that exhibits topological linking at infinity yet has scalar curvature failing to decay at any polynomial rate would disprove the forcing result.

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Extended reading notes

Core claim

On complete non-compact manifolds of weakly bounded geometry, topological linking at infinity forces the scalar curvature to decay at a polynomial rate. This generalizes examples of quadratic decay and, using mu-bubble exhaustions along with analysis of stable minimal hypersurfaces and index theory, yields qualitative obstructions to uniformly positive scalar curvature localized at individual ends.

Load-bearing premise

The manifold has weakly bounded geometry, without which topological linking at infinity may fail to force polynomial scalar curvature decay.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper studies complete non-compact manifolds of positive scalar curvature. Its central result establishes that topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry, generalizing recent examples with quadratic decay. Building on this, the authors develop an obstruction theory for uniformly positive scalar curvature on individual ends by means of μ-bubble exhaustions, stable minimal hypersurfaces, and index-theoretic arguments.

Significance. If the results hold, the work supplies a conceptual topological mechanism that controls curvature decay at infinity and yields qualitative obstructions to positive scalar curvature on ends. The argument relies on uniform control of the second fundamental form and stability operator under the weakly bounded geometry hypothesis, together with a min-max construction that produces a positive lower bound on the decay rate. The manuscript presents a coherent derivation without internal gaps or hidden assumptions.

minor comments (3)
  1. [§1] §1 (Introduction): the precise polynomial decay rate (e.g., the exponent α in |Scal| = O(r^{-α})) should be stated explicitly in the main theorem rather than left as “polynomial.”
  2. [§3] §3 (μ-bubble exhaustions): a brief reminder of the definition of a μ-bubble and the stability operator would help readers who are not specialists in the technique.
  3. [Theorem 1.3] The statement of the obstruction theorem for individual ends should clarify whether the index-theoretic vanishing holds for all ends or only for those satisfying the linking condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. The recommendation of minor revision is noted; we will prepare a revised version incorporating any editorial or minor clarifications that may be needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation proceeds from the topological linking-at-infinity hypothesis on manifolds of weakly bounded geometry, using standard mu-bubble exhaustion constructions to produce stable minimal hypersurfaces whose index-theoretic properties then yield the polynomial scalar-curvature decay and endwise obstructions. The weakly bounded geometry assumption supplies uniform control on the second fundamental form and stability operator, which is an independent analytic hypothesis rather than a quantity fitted to the target decay rate. No equation reduces to its own input by construction, no parameter is renamed as a prediction, and no load-bearing step rests on a self-citation chain. The argument is therefore self-contained against external analytic tools.

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

The claims rest on the domain assumption of weakly bounded geometry and standard tools from minimal hypersurface theory and index theory; no free parameters or invented entities are mentioned.

assumptions (1)
  • domain assumption The manifold has weakly bounded geometry
    Invoked explicitly for the polynomial decay result under topological linking at infinity.

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Cite this review

Pith. "Pith review of Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds." pith.science (2026). https://pith.science/paper/2604.06547

@misc{pith2026260406547,
  author       = {Pith},
  title        = {Pith review of: Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.06547}},
  note         = {Machine review of arXiv:2604.06547}
}
abstract

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry. This result provides a conceptual generalization of recently discovered examples of metrics with quadratic scalar curvature decay. Building on this decay mechanism, we develop an obstruction theory localized at the ends of non-compact manifolds. Using $\mu$--bubble exhaustions together with the analysis of stable minimal hypersurfaces and index theory, we obtain qualitative obstructions to uniformly positive scalar curvature on individual ends.

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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. Some constructions of uniformly positive scalar curvature metrics on open manifolds

    math.DG 2026-06 unverdicted novelty 6.0 of 10

    Open manifolds with proper Morse functions lacking high-index critical points or with mean convex exhaustions admit uniformly positive scalar curvature metrics.

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Works this paper leans on

7 extracted references · 7 canonical work pages · cited by 1 Pith paper

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