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

Shunichiro Orikasa

arxiv: 2604.06547 · v1 · submitted 2026-04-08 · 🧮 math.DG · math.AP· math.GT

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

Shunichiro Orikasa This is my paper

Pith reviewed 2026-05-10 18:21 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.GT

keywords non-compact manifoldsscalar curvaturepositive scalar curvaturelinking at infinitycurvature decaymu-bubblesminimal hypersurfacesindex theory

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The pith

Topological linking at infinity forces polynomial decay of scalar curvature on non-compact manifolds of weakly bounded geometry.

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

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.

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.

What carries the argument

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

Load-bearing premise

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

read the original 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.

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.

Desk Editor's Note private letter to a colleague

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

The main point is that topological linking at infinity implies polynomial decay of scalar curvature on complete non-compact manifolds with weakly bounded geometry, and this mechanism produces qualitative obstructions to uniformly positive scalar curvature on separate ends. The paper treats linking as a new conceptual device that extends the quadratic-decay examples already in the literature and localizes the obstruction at infinity rather than globally. The argument proceeds by constructing mu-bubble exhaustions, producing stable minimal hypersurfaces whose index properties then obstruct positive curvature, with the weakly bounded geometry supplying the uniform control needed for the second fundamental form and stability operator estimates. The stress-test confirms the estimates close without hidden gaps or circularity once the geometry hypothesis is in place. That control is stated explicitly and used consistently, so the central claims hold up on their own terms. The main limitation is the weakly bounded geometry assumption itself; without it the min-max construction and index bounds lose their uniformity, but the paper does not pretend otherwise. The citation pattern looks standard for the subfield and the new tool is presented as a direct generalization rather than a routine extension. Readers working on positive scalar curvature for non-compact manifolds or on asymptotic geometry will find the localized obstruction theory useful. The work is coherent enough and the ideas fresh enough that it should go to a serious referee rather than a desk reject.

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 (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.”
[§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.
[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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (SphereAdmitsCircleLinking D ↔ D=3) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem A (Theorem 4.1): ... Lk(F, α)≠0 ... min R_g ≤ C (Dil_1(π_ρ)/ρ)^2 on π^{-1}(B^3(ρ))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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