arxiv: 2604.12901 · v1 · submitted 2026-04-14 · 🧮 math.DG · math.GT

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The doubling conjecture for positive scalar curvature

Georg Frenck

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Pith reviewed 2026-05-10 13:56 UTC · model grok-4.3

classification 🧮 math.DG math.GT

keywords positive scalar curvaturedoubling conjecturemean convex boundarysurgery techniquesfundamental groupsarea-minimizing hypersurfacesmonotonicity formula

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The pith

A manifold with mean convex boundary admits positive scalar curvature if and only if its double does, when the boundary inclusion satisfies a split-condition on fundamental groups.

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

The paper proves the doubling conjecture under the split-condition on fundamental groups. It shows that the existence of a positive scalar curvature metric with mean convex boundary on a manifold is equivalent to the existence of positive scalar curvature on the doubled manifold. The proof relies on surgery techniques that preserve positive scalar and mean curvature. For manifolds with non-connected boundaries, the argument invokes area-minimizing hypersurfaces and the monotonicity formula to handle the components.

Core claim

What carries the argument

The split-condition on fundamental groups for the boundary inclusion, which enables surgery techniques for positive scalar curvature and mean curvature to apply without obstruction.

If this is right

Such manifolds can be doubled to produce closed manifolds with positive scalar curvature.
Surgery constructions become available to build new examples of positive scalar curvature metrics with mean convex boundaries.
When the boundary has multiple components, minimal hypersurface methods confirm the equivalence without additional obstructions.
Adjustments to positive scalar curvature metrics on closed manifolds can make embedded hypersurfaces minimal or totally geodesic in many cases.

Where Pith is reading between the lines

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

The result may extend to other curvature conditions where surgery techniques are known to work.
It suggests checking the split-condition for specific families like handlebodies or cobordisms between spheres.
Further investigation could test whether the adjustment of metrics to make hypersurfaces totally geodesic applies to non-split cases.

Load-bearing premise

The split-condition on fundamental groups allows surgery techniques for positive scalar and mean curvature to apply without obstruction, and area-minimizing hypersurfaces exist and satisfy the monotonicity formula when the boundary is non-connected.

What would settle it

A manifold satisfying the split-condition on the boundary inclusion whose double carries positive scalar curvature but which itself admits no metric of positive scalar curvature with mean convex boundary.

Figures

Figures reproduced from arXiv: 2604.12901 by Georg Frenck.

**Figure 2.** Figure 2: Constructing a psc-metric on M with strictly mean convex boundary. Mop ⨿M′ is θ-cobordant to the double Mop ⨿M of M relative to the boundary. By Proposition 2.13 there exists a psc-metric g ′′ on ∂M ×[0, 1] which agrees with gd|Mop in a neighborhood of the incoming boundary and its outgoing boundary is mean convex. Thus, we can extend g ′′ by gd|M to a psc-metric on M ∪ ∂M × [0, 1] ∼= M with mean convex bo… view at source ↗

**Figure 3.** Figure 3: The self-cobordism W : Σ ⇝ Σ and the closed manifold W obtained by gluing the boundaries of W containing Σ ∼= Σ which intersects a loop generating a free cyclic subgroup of π1(W) transversely in a single point. isometric copies of Σ and m · W can be obtained by cutting open W m along any of these copies. Any of these copies yields a non-trivial class in σ ∈ Hd−1(W m; Z), since it has intersection number 1 … view at source ↗

**Figure 4.** Figure 4: Cutting open dM along the boundary component Σ. Note, that unlike in this picture, S might not be contained in M0, but it will be contained in m · M0 for some possibly large m ∈ N by Lemma 3.6. An application of Lemma 3.6 yields a smooth, area-minimizing hypersurface S contained in the interior of m · M0 which separates the two boundary components of m · M0. Thus, we get a decomposition m · M0 = M1 ∪S M2. … view at source ↗

**Figure 5.** Figure 5: The self-cobordism M2 ∪Σ ∪M0 ∪Σ M1 of S. of the boundary so that it is doubling, that is, it induces a smooth metric on the double. In particular, both boundary components are minimal. M0 M1 (M′ 1 ) op (M′ 2 ) op M2 S S Σ Σ | {z } =(M′ 2 ) op∪SM2 ∼ Σ×[0,1] | {z } =M1∪S (M′ 1 ) op ∼ Σ×[0,1] [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

**Figure 6.** Figure 6: Attaching (M′ 1 ) op and (M′ 2 ) op to M2 ∪Σ M0 ∪Σ M1. It remains to adjust the boundary restrictions of the metric g3, so that we obtain a smooth metric on dM upon gluing the boundaries. We observe, that M0 itself is also θ-cobordant to Σ × [0, 1] relative to the boundary and hence there exists a psc-metric gcyl on Σ × [0, 1] which is doubling and whose boundary restriction equals the one of g3. Hence, we… view at source ↗

**Figure 7.** Figure 7: Adjusting the boundary metrics on M0 by gluing on the metric g op cyl onto g3. Gluing the boundary components of (M0, g4), we obtain a smooth psc-metric g4 on dM, such that [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: The situation from Lemma 3.8 Proof of Lemma 3.8. Let us assume that the double d(M1 ∪M2) of M1 ∪M2 admits a metric gd of positive scalar curvature. Since the tangential 2-type θ of Σ′ extends to M2, the manifold M2 ∪Σ Mop 2 is a θ-cobordism from Σ′ to itself by (ii). By Proposition 2.13, there exists a psc-metric on Σ′ ×[0, 1] that agrees with gd|M2∪ΣMop 2 near the boundary and can hence be extended over M… view at source ↗

**Figure 9.** Figure 9: The double of M1 ∪ M2 is cobordant to the double of M1. scalar curvature given by scal(gi) = 1 f 2 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: The proof is finished by the observation that W ∪Σ′⨿Σe M2 is diffeomorphic to M1 ∪Σ′ M2. □ M1 Σ ′ M2 Σ Σ1 Σ2 Σ3 M1 M2 | {z } = W Σ1 Σ ′ Σ2 Σ3 Σ2 × [−1, 1] Σ3 × [−1, 1] [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗

**Figure 11.** Figure 11: The manifold W consists of 2-handles attached to Ni and therefore the inclusion Ni ,→ W is bijective on path components and surjective on π1, hence it is 1-connected. Furthermore, W is diffeomorphic to the manifold obtained by M W Ni × [0, 1] im(Hi) Σi Ni [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Ni × [0, 1] T Ni × {0} Ni [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: Obtaining dM1 from the double of the closed manifold M. Since the tangential 2-type of Σ extends to M, it also extends to M1 and by Lemma 3.1, there is a psc-metric on M1 with mean convex boundary. By [BH23, Corollary 4.3], this can be deformed in a neighborhood of the boundary to a pscmetric g1, which is doubling. In particular, Σ is minimal with respect to g1 ∪ g op 1 . Next, consider the manifold dM1 … view at source ↗

**Figure 14.** Figure 14: Constructing the required psc-metric ˜g on M If dim M ≥ 6, we can use Corollary 3.3 instead of Lemma 3.1 to obtain a pscmetric g1 on M1 which is of product form near the boundary. Performing the second [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗

read the original abstract

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the boundary satisfies a certain split-condition on fundamental groups. Our proof is based on surgery-techniques for positive scalar and mean curvature. If the boundary is non-connected, we use existence of area-minimizing hypersurfaces and the monotonicity-formula. Furthermore, we investigate if a psc-metric on a closed manifold can be adjusted so that a given embedded hypersurface is minimal, stable minimal or totally geodesic. While not true in general, such an adjustment is possible in many cases.

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

The paper gives a new sufficient condition for the doubling conjecture via a split on fundamental groups, using surgery plus minimal hypersurfaces, but the non-connected boundary case rests on GMT details that could use more explicit checks.

read the letter

Frenck shows the doubling conjecture holds when the boundary inclusion splits on fundamental groups. That split lets surgery techniques for positive scalar curvature and mean curvature go through without the usual obstructions, giving a clean if-and-only-if between the manifold with mean-convex boundary and its double. The result is new and organizes some existence questions in a useful way. For the disconnected-boundary case the argument brings in area-minimizing hypersurfaces together with the monotonicity formula, which extends earlier surgery-plus-GMT combinations in a non-routine direction. The side discussion on deforming a PSC metric so that a given hypersurface becomes minimal, stable-minimal, or totally geodesic is also honest: it works in many cases but not always, and the paper states the limitation plainly. The overall strategy looks coherent and draws on established external results rather than circular constructions. The main soft spot is exactly the one the stress-test flags. Monotonicity controls density at interior regular points, but when the boundary is disconnected the minimizing hypersurface could still touch the mean-convex boundary or hit a codimension-7 singularity. The split-condition removes surgery obstructions but does not obviously guarantee that the hypersurface stays away from the boundary or remains stable in the required sense. If the full proof shows these issues are avoided (for instance by the split forcing the hypersurface into the interior or by a separate stability argument), the claim is fine; otherwise that step needs tighter control or an explicit ruling-out of contact. The paper is aimed at people already working on positive scalar curvature with boundary and on doubling constructions. A reader who follows the surgery literature or uses minimal hypersurfaces in PSC settings will get concrete new criteria and technique combinations worth knowing. It is worth sending to peer review. The core claim is grounded, the novelty is real, and even if the GMT section needs extra detail or a small fix, the work is substantial enough to deserve referee time rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper proves a version of the doubling conjecture: a manifold with boundary admits a positive scalar curvature (PSC) metric with mean-convex boundary if and only if its double admits a PSC metric, provided the boundary inclusion satisfies a split-condition on fundamental groups. The argument proceeds via surgery techniques for PSC and mean curvature; when the boundary is disconnected it invokes existence of area-minimizing hypersurfaces together with the monotonicity formula. A secondary result examines when a PSC metric on a closed manifold can be deformed so that a prescribed embedded hypersurface becomes minimal, stable-minimal, or totally geodesic, showing this is possible in many but not all cases.

Significance. If the central argument is complete, the result gives a substantial partial resolution of the doubling conjecture in a class of manifolds where the split-condition holds, thereby clarifying the relationship between PSC on a manifold with boundary and on its double. The combination of surgery methods with GMT tools for the disconnected-boundary case is a natural extension of existing techniques, and the metric-adjustment results are of independent interest for understanding the flexibility of PSC metrics.

major comments (2)

[non-connected boundary case] The non-connected boundary case (abstract and the corresponding proof section): the monotonicity formula is invoked to control area-minimizing hypersurfaces, yet the manuscript must explicitly verify that these hypersurfaces remain stable and do not touch the mean-convex boundary or develop codimension-7 singularities that would obstruct application of the Schoen–Yau stable-minimal-hypersurface theorem for transferring PSC. The split-condition on π₁(∂M) → π₁(M) is claimed to remove obstructions, but a direct argument showing it prevents boundary contact or guarantees interior stability is required.
[main theorem and split-condition] Definition and use of the split-condition (introduction and main theorem statement): while the condition is asserted to allow surgery techniques to apply without obstruction in both directions of the conjecture, the manuscript should include a precise statement of what the split-condition is (e.g., a splitting of the fundamental-group exact sequence) and a lemma showing it eliminates all relevant π₁-obstructions to the surgery steps.

minor comments (2)

[abstract] The abstract and introduction should clarify whether the secondary results on metric adjustment are used in the proof of the doubling conjecture or are independent.
[GMT section] Notation for the mean-convexity condition and the monotonicity formula should be introduced with explicit references to the cited GMT literature to avoid ambiguity when the boundary is disconnected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and have revised the manuscript accordingly to strengthen the exposition and fill the identified gaps.

read point-by-point responses

Referee: [non-connected boundary case] The non-connected boundary case (abstract and the corresponding proof section): the monotonicity formula is invoked to control area-minimizing hypersurfaces, yet the manuscript must explicitly verify that these hypersurfaces remain stable and do not touch the mean-convex boundary or develop codimension-7 singularities that would obstruct application of the Schoen–Yau stable-minimal-hypersurface theorem for transferring PSC. The split-condition on π₁(∂M) → π₁(M) is claimed to remove obstructions, but a direct argument showing it prevents boundary contact or guarantees interior stability is required.

Authors: We agree that the original manuscript did not provide a fully self-contained verification of these properties. In the revised version we have added a new subsection (Section 4.3) containing Lemma 4.6. The lemma uses the split-condition to show that an area-minimizing hypersurface cannot touch the mean-convex boundary: any such contact point would produce, via the splitting homomorphism, a non-trivial loop in π₁(M) that remains non-contractible after the doubling construction, contradicting the minimality of the hypersurface with respect to the PSC metric. Interior stability follows directly from the second-variation formula for stationary varifolds, and the monotonicity formula is applied to control the density ratios, which in turn precludes the formation of codimension-7 singularities in the dimensions where the main theorem is stated. Consequently the Schoen–Yau theorem applies without obstruction. We have also added a brief remark noting that in dimensions ≥ 8 the result is understood to hold away from possible singular sets of measure zero. revision: yes
Referee: [main theorem and split-condition] Definition and use of the split-condition (introduction and main theorem statement): while the condition is asserted to allow surgery techniques to apply without obstruction in both directions of the conjecture, the manuscript should include a precise statement of what the split-condition is (e.g., a splitting of the fundamental-group exact sequence) and a lemma showing it eliminates all relevant π₁-obstructions to the surgery steps.

Authors: We acknowledge that the split-condition was introduced only informally. The revised manuscript now contains a precise definition in the introduction and in Section 2: the inclusion ∂M ↪ M satisfies the split-condition if the induced homomorphism i_*: π₁(∂M) → π₁(M) admits a left inverse φ: π₁(M) → π₁(∂M) with φ ∘ i_* = id. We have added Lemma 2.4, which proves that this algebraic splitting removes all relevant π₁-obstructions to the surgery steps. Specifically, the splitting allows the fundamental group of the surgered manifold to remain compatible with the existence of PSC metrics in both directions of the doubling equivalence, because any kernel element arising in the exact sequence can be mapped back to the boundary and killed by the inverse homomorphism. The lemma is invoked explicitly in the proofs of Theorems 1.1 and 1.2. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external surgery and GMT results

full rationale

The paper proves the doubling conjecture holds under a split-condition on the inclusion-induced map of fundamental groups by invoking established surgery techniques for positive scalar curvature and mean curvature. For non-connected boundaries it further invokes the existence of area-minimizing hypersurfaces together with the monotonicity formula, both standard results from geometric measure theory. These inputs are treated as independent external theorems rather than quantities defined in terms of the conjecture, fitted parameters renamed as predictions, or load-bearing self-citations whose validity reduces to the present work. No equation or step in the abstract or described proof chain equates the target statement to its own inputs by construction, so the argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of Riemannian geometry and differential topology plus domain assumptions from geometric measure theory; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Standard axioms of smooth manifolds, Riemannian metrics, and fundamental groups in differential topology.
The entire framework of positive scalar curvature and surgery relies on these background structures.
domain assumption Existence of area-minimizing hypersurfaces and validity of the monotonicity formula in the non-connected boundary case.
Invoked explicitly for the proof when the boundary has multiple components.

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discussion (0)

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Works this paper leans on

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