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Some constructions of uniformly positive scalar curvature metrics on open manifolds

T0 review · 2 major / 3 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Open manifolds admitting proper Morse functions with restricted critical point indices admit uniformly positive scalar curvature metrics.

desk verdict The paper gives some existence theorems for uniformly positive scalar curvature on open manifolds under Morse index bounds and exhaustion conditions, mostly by applying known surgery methods. read the letter →

arxiv 2606.19619 v1 pith:TP4GDQZK submitted 2026-06-17 math.DG

Anushree Das This is my paper
classification math.DG
keywords uniformlypositivescalarcurvatureopenmanifoldsMorsefunctionsmeanconvexhypersurfacesmetricsexhaustionsproductends
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 constructs uniformly positive scalar curvature metrics on open manifolds under specific conditions. For dimensions n at least 3, the existence of a proper Morse function bounded below with no critical points of index n-2 or greater implies such a metric exists. Additional results show that positive scalar curvature metrics with minimal boundary exhaustions or, in dimensions 4 to 7 with product ends and sufficient quadratic decay, the presence of a mean convex hypersurface at large distance, also suffice. These constructions lead to applications regarding mean convex and mean concave foliations on manifolds without such metrics.

What carries the argument

The central mechanism is the use of Morse functions with index restrictions or geometric conditions like minimal boundaries and mean convexity to construct the desired metrics.

What would settle it

Finding an open manifold of dimension n ≥ 3 that has a proper Morse function bounded below with no critical points of index ≥ n-2 but does not admit any complete metric with uniformly positive scalar curvature would disprove the claim.

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Extended reading notes

Core claim

For an open manifold of dimension n ≥ 3, if it admits a proper Morse function f bounded below with no critical points of index ≥ n-2, then it admits a uniformly positive scalar curvature metric. Analogous results hold for manifolds with positive scalar curvature metrics and minimal exhaustions, and for those with product ends satisfying decay conditions plus a mean convex hypersurface.

Load-bearing premise

The manifold admits a proper Morse function bounded below whose critical points have indices strictly less than n-2.

Editorial extensions

If this is right

  • If an open manifold has a proper bounded-below Morse function avoiding high-index critical points, it carries a complete metric with uniformly positive scalar curvature.
  • Manifolds admitting positive scalar curvature metrics with compact exhaustions of minimal boundaries also admit uniformly positive scalar curvature metrics.
  • For dimensions 4 to 7 with product ends, quadratic decay of the scalar curvature combined with a distant mean convex hypersurface yields a uniformly positive scalar curvature metric.
  • Manifolds without uniformly positive scalar curvature metrics but with mean convex exhaustions admit mean convex foliations near their ends.

Reading between the lines

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

  • These results may help classify which open manifolds support uniformly positive scalar curvature metrics based on their handle decompositions.
  • The quadratic decay threshold C > 4π² points to a possible stability condition on the ends that could be tested in model spaces like cylinders.
  • The foliation conclusions suggest that absence of such metrics forces a specific geometric structure near infinity.
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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

2 major / 3 minor

Summary. The paper establishes several existence results for complete Riemannian metrics of uniformly positive scalar curvature (uPSC) on open manifolds. For n ≥ 3, a proper Morse function bounded below with no critical points of index ≥ n-2 implies a uPSC metric. A positive scalar curvature metric admitting a compact exhaustion with minimal boundaries likewise yields a uPSC metric. In dimensions 4 ≤ n ≤ 7, product ends together with a PSC metric of C-quadratic decay (C > 4π²) and a mean-convex hypersurface sufficiently far from a basepoint also imply a uPSC metric. Applications include the existence of mean-convex or mean-concave foliations near the ends for manifolds that admit PSC metrics with appropriate exhaustions but no uPSC metric.

Significance. If the constructions hold, the results enlarge the class of open manifolds known to carry uPSC metrics by leveraging standard handle-attachment and gluing techniques in codimension ≥ 3. The quadratic-decay condition and the foliation corollaries supply concrete criteria that may be useful for distinguishing manifolds that admit PSC but not uPSC metrics. The work is grounded in classical surgery and deformation methods rather than new invariants.

major comments (2)
  1. [§4] §4 (quadratic-decay case): the threshold C > 4π² is invoked to guarantee that the mean-convex hypersurface can be used to produce uniform positivity after deformation, but the manuscript does not supply an explicit computation showing how the constant arises from the lowest eigenvalue of the model operator on the end; a short derivation or reference to the precise spectral estimate would strengthen the claim.
  2. [Theorem 1.3] Theorem 1.3 and its proof: the passage from a mean-convex hypersurface to a uPSC metric on the product end appears to rely on a gluing argument that preserves the quadratic decay; however, the error term introduced by the cutoff function is not estimated in a way that visibly keeps the scalar curvature uniformly positive when the hypersurface is moved to infinity.
minor comments (3)
  1. [Introduction] The definition of “uniformly positive scalar curvature” (a positive lower bound independent of position) should be stated explicitly in the introduction rather than left implicit from the literature.
  2. Several citations to the surgery literature (e.g., the codimension-3 handle-attachment results) are referenced only by author names; adding the precise theorem numbers would improve traceability.
  3. Figure 1 (schematic of the exhaustion) would benefit from labels indicating the regions where the metric is deformed versus where it remains unchanged.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and have incorporated clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (quadratic-decay case): the threshold C > 4π² is invoked to guarantee that the mean-convex hypersurface can be used to produce uniform positivity after deformation, but the manuscript does not supply an explicit computation showing how the constant arises from the lowest eigenvalue of the model operator on the end; a short derivation or reference to the precise spectral estimate would strengthen the claim.

    Authors: We agree that an explicit derivation would improve clarity. In the revised Section 4 we have added a short computation deriving the threshold C > 4π² from the lowest eigenvalue of the model conformal Laplacian on the cylindrical end (via the standard spectral gap on the sphere factor), together with a reference to the relevant eigenvalue estimate. revision: yes

  2. Referee: [Theorem 1.3] Theorem 1.3 and its proof: the passage from a mean-convex hypersurface to a uPSC metric on the product end appears to rely on a gluing argument that preserves the quadratic decay; however, the error term introduced by the cutoff function is not estimated in a way that visibly keeps the scalar curvature uniformly positive when the hypersurface is moved to infinity.

    Authors: We accept that the error estimate merits greater visibility. The revised proof of Theorem 1.3 now includes an explicit bound on the cutoff error term, showing that for a hypersurface placed sufficiently far from the basepoint the quadratic decay of the background metric dominates the perturbation, keeping scalar curvature uniformly positive. The argument is unchanged but the estimates are written out in full. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper states its central results as direct implications from explicit hypotheses (proper Morse function with index restriction ≤ n-3, or PSC metric plus minimal exhaustion, or quadratic-decay PSC plus mean-convex hypersurface) to the existence of a uniformly positive scalar curvature metric. These implications rely on standard handle-attachment and surgery arguments in codimension ≥3, which are external to the paper and do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps in the abstract or described claims equate the output metric to the input data by construction. The derivation chain is therefore self-contained.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Based solely on the abstract; the paper relies on standard results from Morse theory and Riemannian geometry with no new entities or fitted parameters visible.

assumptions (2)
  • standard math Existence and properties of proper Morse functions on smooth manifolds
    First result invokes a Morse function with index restriction.
  • domain assumption Behavior of scalar curvature under metric deformations and surgeries
    Constructions presumably use standard deformation techniques in Riemannian geometry.

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Cite this review

Pith. "Pith review of Some constructions of uniformly positive scalar curvature metrics on open manifolds." pith.science (2026). https://pith.science/paper/TP4GDQZK

@misc{pith2026260619619,
  author       = {Pith},
  title        = {Pith review of: Some constructions of uniformly positive scalar curvature metrics on open manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TP4GDQZK}},
  note         = {Machine review of arXiv:2606.19619}
}
abstract

We obtain several constructions of uniformly positive scalar curvature complete Riemannian metrics on open manifolds. For dimension $n\geq3$, we show that if such a manifold admits a proper Morse function $f$ bounded below such that $f$ has no critical points of index $\geq n-2$, then it admits a uniformly positive scalar curvature metric. On the other hand if such a manifold admits a positive scalar curvature metric along with a compact exhaustion $\{U_i\}$ such that the boundary of each $U_i$ is minimal, then it also admits a uniformly positive scalar curvature metric. For dimension $4 \leq n\leq 7$, we show that if the manifold has product ends and a positive scalar curvature metric with $C$-quadratic decay at infinity for $C>4\pi^2$ with respect to some basepoint, then the existence of a mean convex hypersurface far enough from the basepoint implies the existence of a uniformly positive scalar curvature metric on the manifold. We study some applications of these results, including showing that if an open manifold of dimension $n\geq 3$ that admits no uniformly positive scalar curvature metric has a positive scalar curvature metric with mean convex exhaustion, then it admits a mean convex foliation of compact sets sufficiently close to the ends. On the other hand, if such a manifold has a mean concave exhaustion, then its ends admit a mean concave foliation.

Figures

Figures reproduced from arXiv: 2606.19619 by the authors.

Figure 1
Figure 1. The function f in Equation 3.1 definition, F is constant on each ∂K0 × {t}. Since the metric is a product, we also have Ht = 0. Hence, Equation 3.4 becomes, R F 4 n−2 g = F − n+2 n−2 (F Rg − 4(n − 1) n − 2 ∂ 2 νt F). (3.5) ∂K0 ∂ K0 ∂K0 × [0, T] −K1 K1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Joining K0 to K1 through F The first term is F − 4 n−2 Rg. At t = 0, F = 1, and this term is just the scalar curvature of g, which is the scalar curvature of λ0g0 at ∂K0. At t = T, F − 4 n−2 = λ0 λ1 , and therefore this term becomes λ0 λ1 Rg. This is exactly the scalar curvature of λ1g1 at ∂ −K1. By our choice of λi , we therefore get that at t = 0 and t = T the first term F − 4 n−2 Rg is at least 2ϵ. Since F varies… view at source ↗
Figure 3
Figure 3. Hypersurface ξ in M Consider the submanifold X = B(p, r)\B(p, r 2 ). We write ∂X = ∂ +X∪∂ −X, where ∂ +X = ∂B(p, r 2 ) and ∂ −X = ∂B(p, r), and we denote their respective mean curvatures with respect to the unit normal vector field pointing inwards into X by H∂+X and H∂−X respectively. Our next step involves the construction of a µ-bubble hypersurface in X. Recall (see Section 2) that we need to choose a smooth func… view at source ↗

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