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Positive curvature conditions on contractible manifolds

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Pith's one-line read Uniformly positive scalar curvature forces the interior of a suitably connected contractible 5-manifold to be diffeomorphic to $\mathbb{R}^5$, and stronger boundary-curvature conditions force compact contractible manifolds to be…

desk verdict Theorem A is a real new result and the strategy is sound, but the n=3 part of Theorem B(ii) has an opaque, undefined step that needs repair; overall the paper deserves peer review. read the letter →

arxiv 2507.15719 v1 pith:ZMIT2HI5 submitted 2025-07-21 math.DG

Paul Sweeney Jr This is my paper
classification math.DG MSC 53C2153C2457K40
keywords positivescalarcurvaturecontractiblemanifoldsmu-bubblesisotropicmeanconvexboundaryRiccipinchinghomologyspheresMazur
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 asks which curvature conditions force an open contractible manifold to be Euclidean space and a compact contractible manifold with boundary to be a standard disk. Its first result shows that if a 5-manifold is the interior of a compact contractible manifold with boundary $X$ satisfying $\pi_3(X,\partial X)=0$, and it admits a complete metric of uniformly positive scalar curvature, then it is diffeomorphic to $\mathbb{R}^5$. For compact manifolds with boundary, the paper shows that positive scalar curvature with mean convex boundary is too weak: many non-disk contractible manifolds admit such metrics. It then proves that two stronger conditions do distinguish the disk—positive isotropic curvature (a positivity condition on the Riemann curvature over isotropic two-planes) with 2-convex boundary in dimensions $4$ and $n\geq 12$, and a Ricci-pinching condition with convex boundary in dimensions $3$ and $4$—together with a related boundary-convexity condition giving disk conclusions in dimensions $3,4,5$. The proofs use $\mu$-bubbles to turn interior curvature into positively curved hypersurfaces near the boundary, then feed that information into known classifications of closed manifolds with positive scalar or isotropic curvature.

What carries the argument

The engine of the proof is the $\mu$-bubble: a minimizer of an area functional with a carefully chosen weight function that tends to $+\infty$ on one boundary component and $-\infty$ on the other. In a manifold of uniformly positive scalar curvature, $\mu$-bubbles produce smooth embedded hypersurfaces that separate boundary components and themselves carry Riemannian metrics of positive scalar curvature. Applied to an exhaustion of the open 5-manifold, this yields hypersurfaces near the boundary with positive scalar curvature; a non-zero-degree projection to the boundary then lets the classification of closed manifolds with positive scalar curvature in dimensions four and five restrict what the boundary can be. For the compact theorems, the same separation mechanism is replaced by direct boundary curvature transfer: condition (C1) is deformed by a positivity-preserving deformation to make the boundary totally geodesic, so the boundary inherits positive isotropic curvature; condition (C2) is shown by the Gauss equations and the standard rearrangement trick for scalar curvature to imply the boundary has positive scalar curvature. The final step in every case is purely topological: a homology-sphere boundary that is covered by a sphere must be simply connected (except for the binary icosahedral group in dimension three), and a simply connected homology sphere is a homotopy sphere, which by known classification theorems yields a disk.

What would settle it

Build a compact, contractible 4-manifold whose boundary is a nontrivial connected sum of Poincaré homology 3-spheres and give it a Ricci-pinched metric with convex boundary; even one such example would contradict the three-dimensional conclusion of Theorem B(ii). A less geometric check is to compute the correction invariant (d-invariant) of that connected sum and verify whether it, together with the cited gauge-theoretic theorem, actually forces the standard 3-sphere as the only boundary.

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Extended reading notes

Core claim

The central discovery is that, in the right topology, uniform positive scalar curvature is a rigidity condition rather than merely a constraint: for a 5-manifold that is the interior of a compact contractible manifold with boundary $X$ and $\pi_3(X,\partial X)=0$, a complete metric of uniformly positive scalar curvature forces the manifold to be diffeomorphic to $\mathbb{R}^5$. The compact analogue is subtler. The paper exhibits, via known constructions, many compact contractible manifolds with boundary that support positive scalar curvature and mean convex boundary, so those hypotheses alone cannot characterize the disk. It then shows that adding a stronger boundary/interior curvature condition does characterize the disk: under condition (C1), positive isotropic curvature with 2-convex boundary, the boundary is diffeomorphic to a sphere in dimensions $n=4$ and $n\geq 12$; under condition (C2), the Ricci pinching $n g \leq \mathrm{Ric} \leq \frac{1}{2}n(n+1)g$ with convex boundary, the boundary is homeomorphic to a sphere for $n=3,4$ (with $\pi_3(X,\partial X)=0$ when $n=4$). From a spherical boundary, h-cobordism and 4-manifold topology results imply the whole manifold is homeomorphic to a disk, and diffeomorphic in several cases. The paper also derives disk conclusions from a boundary-convexity condition (C3) in dimensions $3,4,5$ under the same relative-homotopy hypotheses.

Load-bearing premise

In the three-dimensional case, the proof assumes that a gauge-theoretic theorem, together with the standard correction invariant for homology spheres, rules out every nontrivial connected sum of Poincaré homology spheres as the boundary of a contractible 4-manifold; the paper does not state that theorem or define the symbols in the step where it is used.

Editorial extensions

If this is right

  • If Theorem A is correct, the interior of any compact contractible 5-manifold with boundary satisfying $\pi_3(X,\partial X)=0$ and admitting a complete uniformly positive scalar curvature metric is the standard smooth $\mathbb{R}^5$, so no exotic smooth structure on $\mathbb{R}^5$ can arise from this construction.
  • Under condition (C1), a compact contractible manifold with boundary is homeomorphic to a disk in dimensions 4 and $n\geq 12$, and diffeomorphic when $n\geq 12$, so positive isotropic curvature plus 2-convex boundary is a genuine disk-detecting hypothesis.
  • Under condition (C2), Ricci pinching with convex boundary forces the disk in dimensions 3 and 4, showing that a curvature condition strictly weaker than positive sectional curvature can still single out the disk among contractible manifolds.
  • Corollary C extends the disk conclusion to Wang's boundary-convexity condition (C3) in dimensions 3, 4, and 5, reinforcing that the boundary curvature is what carries the compact rigidity.
  • The contrast between the open and compact cases is sharp: positive scalar curvature plus mean convex boundary does not characterize the disk, since many non-disk contractible manifolds admit such metrics, whereas the interior version with completeness is rigid in dimension 5.

Reading between the lines

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

  • A natural testable extension is to push the same $\mu$-bubble strategy to contractible 6-manifold interiors with complete uniformly positive scalar curvature, where the geometric separation machinery still works; the missing ingredient would be a closed-manifold positive-scalar-curvature classification in dimension six.
  • The paper effectively isolates the boundary homeomorphism type as the place where rigidity happens: if future results produced other homology-sphere boundaries carrying the relevant curvature, the disk conclusions would extend, and the examples show why boundary conditions cannot simply be dropped.
  • The three-dimensional dependence on a gauge-theoretic step suggests a concrete project: a purely four-dimensional proof that a connected sum of Poincaré homology spheres cannot bound a contractible 4-manifold would remove the most delicate assumption in condition (C2).
  • The high-dimensional range $n\geq 12$ in condition (C1) is tied to the currently available classification of closed manifolds with positive isotropic curvature; improved classification in lower dimensions would bring the disk conclusion to those dimensions as well.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper addresses two related questions in positive curvature topology: whether an open contractible manifold with a complete metric of uniformly positive scalar curvature must be Euclidean space, and whether a compact contractible manifold with boundary and stronger curvature conditions must be a disk. Theorem A proves that if M is the interior of a compact contractible 5-manifold X with boundary satisfying π3(X, ∂X)=0, and M admits a complete metric of uniformly positive scalar curvature, then M is diffeomorphic to R^5. Theorem B establishes disk recognition under condition (C1) (positive isotropic curvature and 2-convex boundary) in dimensions n=4 and n≥12, and under condition (C2) (pinched Ricci curvature and convex boundary) in dimensions n=3 and n=4 with an additional topological hypothesis in the latter case. Corollary C applies Wang's condition (C3) to dimensions n=3,4,5. The proofs combine µ-bubble methods, classification results for positive scalar curvature and positive isotropic curvature, Heegaard-Floer correction terms, and algebraic topology of boundaries of contractible manifolds.

Significance. If the results hold, Theorem A is a natural five-dimensional analogue of the Chodosh-Maximo-Mukherjee theorem for open 4-manifolds, and Theorem B provides new positive answers to the disk-recognition question under hypotheses substantially weaker than positive sectional curvature with convex boundary. The paper is well organized and mostly assembles external tools without introducing free parameters or circular reasoning; the use of PIN classification results, µ-bubbles, and the d-invariant is appropriate. The main caveats are that two proof steps are currently incomplete as written: the n=3 case of Theorem B(ii) relies on an informal argument with undefined symbols and an unspecified cited theorem, and Proposition 4.1 omits part of the homology argument needed for odd-dimensional spherical space form summands. These are local and likely repairable, but they affect the rigor of the paper's central claims.

major comments (3)
  1. [Section 4, Proposition 4.2] The n=3 case of Proposition 4.2 is not proved as written. The outline says 'one concludes L = M. Then, by a theorem of Taubes [62], we conclude L = M = 0', but L and M are never defined, and the cited Taubes theorem is not stated. The preceding d-invariant information can at best force equality of the numbers of Poincare homology sphere summands with opposite orientations; it does not by itself rule out a connected sum such as P # (-P). Since Theorem B(ii) for n=3 depends on this step, the author should either replace the outline with a precise citation that proves exactly the needed statement (for instance [19, Prop. 4.2], if it indeed covers this case), or state the Taubes theorem explicitly and define the symbols L and M.
  2. [Section 4, Proposition 4.1] In the odd-n case of Proposition 4.1, the conclusion that each summand Sn/Γj is an integral homology sphere is not justified by the displayed homology computation. The displayed range 2 ≤ i ≤ n-1 omits H1, and the argument that H1(∂X)=0 forces the abelianization of each Γj to be trivial is absent. One must use the connected-sum formula H1(∂X)=⊕ H1(Sn/Γj) together with H1(∂X)=0 from Proposition 3.1 before applying Theorem 3.3. Without this step, the vanishing of J for odd n≥12 is not fully proved.
  3. [Section 3, Corollaries 3.17 and 3.19] Corollary 3.17 is stated as 'Let X^{n+1}, n∈{4,5}, be a compact, contractible n-manifold with boundary', but the notation X^{n+1} and the subsequent proof indicate that X should be an (n+1)-manifold. The same dimensional error appears in Corollary 3.19. Since these statements are used in the proof of Theorem A, the dimensions should be corrected to avoid ambiguity. Additionally, the degree argument in the proof of Corollary 3.17 should explicitly address the case where ∂Ωi has several components; the current sentence 'the restriction π|∂Ωi has non-zero degree' is only implicit and the total degree of the disconnected domain is what is needed.
minor comments (5)
  1. [Section 1] There are several typos: 'nonnegateve' should be 'nonnegative', 'the only open 2 2-manifold' has a duplicated '2', and 'Theorem B' proof begins 'Let X n+1 is a compact'.
  2. [Section 3.2.1, Proposition 3.14] The proof of Proposition 3.14 is labelled a sketch and the displayed function τ2 in (3.2) should be τ+. Since this proposition is a known result of Gromov and is used later, please either provide a complete derivation of the inequalities leading to (3.6) or clearly relegate the proof to [30, Section 3.7] and [19, Prop. 3.10].
  3. [Section 4, Proposition 4.2] The term 'Heegard-Floer' should be 'Heegaard-Floer'. Also, the sentence 'By applying the Heegard-Floer d-invariant [44, Theorem 1.2, Proposition 4.2, Proposition 4.3, Section 8.1, and Proposition 9.9] one concludes L = M' should state which property of the d-invariant is being used and why it gives equality rather than vanishing.
  4. [Section 4, Proposition 4.3] In the n=5 case of Proposition 4.3, after concluding that a finite cover of ∂X is homotopy equivalent to S^5, the proof should explicitly mention that this implies ∂X is covered by S^5 and then apply Theorem 3.3 to conclude π1(∂X)=0 before invoking Milnor's result. The current text skips this step.
  5. [Section 4, proof of Corollary C] The sentence 'Then X homeomorphic to the (n+1)-disk' is missing the verb 'is'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain uses external classification theorems and no fitted parameters; the flagged n=3 gap in Proposition 4.2 is a correctness issue, not a circular step.

full rationale

I walked the derivation chain and found no step in which a predicted or derived conclusion is equivalent by construction to an assumed input, and no load-bearing self-citation. The author does not cite his own prior work; the paper instead relies on independent external results: the Chodosh–Li–Liokumovich classification for sufficiently connected PSC manifolds [18], the Chen–Tang–Zhu and Huang PIC classifications [14, 36], Sjerve's theorem on homology spheres covered by spheres [57], Freedman's homeomorphism classification [24], Milnor's h-cobordism results [43], Stallings' uniqueness of the smooth structure on R^5 [60], Gromov's µ-bubble separation theorem, Perelman's and Hamilton's 3-manifold classifications, and Ozsváth–Szabó d-invariant facts [44]. Theorem A reduces the problem to the boundary being a homotopy 4-sphere and then applies Freedman and Milnor; Theorem B(i) reduces to known PIC classifications; Theorem B(ii) reduces to PSC and Ricci curvature classifications plus topological lemmas; Corollary C reduces to Wang's contractibility criterion and the same external steps. None of these steps assume the target conclusion that X is a disk or that M is R^5. The one flagged weakness is in Proposition 4.2, n=3: the text states 'one concludes L = M. Then, by a theorem of Taubes [62], we conclude L = M = 0' without defining L and M and without stating the applicable Taubes theorem. This is a serious proof gap and a correctness risk, because the d-invariant argument alone does not visibly rule out connected sums such as P # (-P). However, an omitted or underspecified external citation is not circularity: the passage does not exhibit the target conclusion as an input, and no equation reduces to itself. Proportionally, the honest circularity finding is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The ledger is clean: no fitted constants and no invented entities. The central results rest entirely on cited external theorems, most of which are standard in geometric analysis and topology; the only flagged item is the unstated Taubes application.

assumptions (8)
  • standard math Gromov's separation theorem and the existence and regularity of mu-bubbles in dimensions 2 through 6.
    Used to prove Proposition 3.13 and Proposition 3.14, which produce exhaustions of open PSC manifolds by compact domains with positive scalar curvature boundary; the paper provides only a sketch.
  • standard math Chodosh-Li-Liokumovich classification theorem for closed PSC n-manifolds with a nonzero degree map to a sufficiently connected target (Theorem 3.15).
    This is the engine for Corollary 3.17, Theorem A, and parts of Proposition 4.2 and Corollary C.
  • standard math Chen-Tang-Zhu classification in dimension 4 and Huang classification in dimension n>=12 of closed manifolds with positive isotropic curvature.
    Used in Proposition 4.1 to identify the boundary of a PIC manifold as a connected sum of standard summands.
  • standard math Sjerve's theorem: an integral homology sphere covered by a sphere is either simply connected or the Poincare homology 3-sphere.
    Used in Theorem A and Propositions 4.1 and 4.2 to upgrade finite spherical covers to simply connected boundaries.
  • standard math Freedman's classification of 4-manifolds and Milnor-Smale h-cobordism/diffeomorphism results stating that a contractible manifold with spherical boundary is a disk in the relevant dimensions.
    Used at the end of Theorem A, Theorem B, and Corollary C to pass from spherical boundary to the disk conclusion.
  • standard math Perelman's classification of closed 3-manifolds with positive scalar curvature and Hamilton's theorem on 3-manifolds with positive Ricci curvature.
    Used in Proposition 4.2 and Proposition 4.3 for the n=3 cases.
  • standard math Ozsvath-Szabo d-invariants and an unspecified theorem of Taubes used to rule out Poincare sphere summands on the boundary of a contractible 4-manifold.
    Used in Proposition 4.2 and Proposition 4.3 n=3; the application is not written out, which is the main gap.
  • standard math Chow's deformation theorem: a PIC metric with 2-convex boundary can be deformed to a PIC metric with totally geodesic boundary.
    Used in Proposition 4.1 to transfer the PIC condition from the manifold interior to the boundary.

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Pith. "Pith review of Positive curvature conditions on contractible manifolds." pith.science (2026). https://pith.science/paper/ZMIT2HI5

@misc{pith2026250715719,
  author       = {Pith},
  title        = {Pith review of: Positive curvature conditions on contractible manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMIT2HI5}},
  note         = {Machine review of arXiv:2507.15719}
}
read the original abstract

Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that is the interior of a sufficiently connected, compact, contractible 5-manifold with boundary and supports a complete Riemannian metric with uniformly positive scalar curvature is diffeomorphic to Euclidean 5-space. Next, we investigate the analogous question for compact manifolds with boundary: Must a compact, contractible manifold that supports a Riemannian metric with positive scalar curvature and mean convex boundary necessarily be the disk? We present examples demonstrating that this curvature condition alone cannot distinguish the disk; on the other hand, we exhibit stronger curvature conditions that allow us to draw such a conclusion.

Figures

Figures reproduced from arXiv: 2507.15719 by the authors.

Figure 1
Figure 1. Schematic picture of a µ-bubble The existence and regularity of a minimizer of A among all Caccioppoli sets is claimed by Gromov [30, Section 5.2] and rigorously carried out by Zhu [70, Proposition 2.1] (cf. [17, Proposition 12]) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Graphs of τ− and τ+ Define now ρ(x) = (1 − δ)τ−(d−(x)) + δτ+(d+(x)). (3.3) Therefore, ρ has the following properties (see [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Graph of ρ Finally, define the smooth function h : X → R h(x) := tan ρ(x) − 1 2C C π ! (3.4) and note that h → ±∞ on ∂±X, respectively (see [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A graph of h With respect to this h, we can find a µ-bubble Ω with boundary ∂Ω = Σ by using Proposition 3.9. By Lemma 3.12, for every ψ ∈ C∞ c (Σ), Z Σ [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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Cited by 1 Pith paper

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  1. Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay

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    An orientable 3-manifold with positive scalar curvature decaying at rate C > 2/3 must be a connected sum of spherical manifolds and S^2 x S^1 pieces, and the threshold 2/3 is optimal.

Reference graph

Works this paper leans on

70 extracted references · 67 canonical work pages · cited by 1 Pith paper

  1. [62]

    Gauge theory on asymptotically periodic 4-manifolds

    Clifford Taubes. “Gauge theory on asymptotically periodic 4-manifolds”. In: J. Dif- ferential Geom. 25.3 (1987), pp. 363–430

    work page 1987

  2. [19]

    Complete Riemannian 4- manifolds with uniformly positive scalar curvature

    Otis Chodosh, Davi Maximo, and Anubhav Mukherjee. Complete Riemannian 4- manifolds with uniformly positive scalar curvature . 2024. arXiv: 2407.05574

    arXiv 2024

  3. [1]

    Complete 3-manifolds of positive scalar curvature with quadratic decay

    Florent Balacheff, Teo Gil Moreno de Mora Sard` a, and St´ ephane Sabourau. “Complete 3-manifolds of positive scalar curvature with quadratic decay”. In: Math. Ann. (2025)

    work page 2025

  4. [2]

    Boundary conditions for scalar curvature

    Christian B¨ ar and Bernhard Hanke. “Boundary conditions for scalar curvature”. In: Perspectives in scalar curvature. Vol. 2 . World Sci. Publ., Hackensack, NJ, 2023, pp. 325–377

    work page 2023

  5. [3]

    Ricci flow on open 3-manifolds and positive scalar curvature

    Laurent Bessi` eres, G´ erard Besson, and Sylvain Maillot. “Ricci flow on open 3-manifolds and positive scalar curvature”. In: Geom. Topol. 15.2 (2011), pp. 927–975

    work page 2011

  6. [4]

    Deform- ing 3-manifolds of bounded geometry and uniformly positive scalar curvature

    Laurent Bessi` eres, G´ erard Besson, Sylvain Maillot, and Fernando Marques. “Deform- ing 3-manifolds of bounded geometry and uniformly positive scalar curvature”. In: J. Eur. Math. Soc. (JEMS) 23.1 (2021), pp. 153–184

    work page 2021

  7. [5]

    Ricci flow and the sphere theorem

    Simon Brendle. Ricci flow and the sphere theorem . Vol. 111. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2010, pp. viii+176

    work page 2010

  8. [6]

    Ricci flow with surgery on manifolds with positive isotropic curva- ture

    Simon Brendle. “Ricci flow with surgery on manifolds with positive isotropic curva- ture”. In: Ann. of Math. (2) 190.2 (2019), pp. 465–559

    work page 2019

Show all 70 references
  1. [7]

    Quasi-isometries and ends of groups

    Stephen Brick. “Quasi-isometries and ends of groups”. In: J. Pure Appl. Algebra 86.1 (1993), pp. 23–33

    1993

  2. [8]

    Metrics of positive Ricci curvature on the connected sums of prod- ucts with arbitrarily many spheres

    Bradley Burdick. “Metrics of positive Ricci curvature on the connected sums of prod- ucts with arbitrarily many spheres”. In:Ann. Global Anal. Geom. 58.4 (2020), pp. 433– 476

    2020

  3. [9]

    Ricci-positive metrics on connected sums of projective spaces

    Bradley Burdick. “Ricci-positive metrics on connected sums of projective spaces”. In: Differential Geom. Appl. 62 (2019), pp. 212–233

    2019

  4. [10]

    Constrained deformations of positive scalar cur- vature metrics

    Alessandro Carlotto and Chao Li. “Constrained deformations of positive scalar cur- vature metrics”. In: J. Differential Geom. 126.2 (2024), pp. 475–554

    2024

  5. [11]

    Constrained deformations of positive scalar cur- vature metrics, II

    Alessandro Carlotto and Chao Li. “Constrained deformations of positive scalar cur- vature metrics, II”. In: Comm. Pure Appl. Math. 77.1 (2024), pp. 795–862

    2024

  6. [12]

    Taming 3-manifolds using scalar curvature

    Stanley Chang, Shmuel Weinberger, and Guoliang Yu. “Taming 3-manifolds using scalar curvature”. In: Geom. Dedicata 148 (2010), pp. 3–14

    2010

  7. [13]

    On the structure of complete manifolds of nonneg- ative curvature

    Jeff Cheeger and Detlef Gromoll. “On the structure of complete manifolds of nonneg- ative curvature”. In: Ann. of Math. (2) 96 (1972), pp. 413–443. REFERENCES 19

    1972

  8. [14]

    Complete classification of com- pact four-manifolds with positive isotropic curvature

    Bing-Long Chen, Siu-Hung Tang, and Xi-Ping Zhu. “Complete classification of com- pact four-manifolds with positive isotropic curvature”. In: J. Differential Geom. 91.1 (2012), pp. 41–80

    2012

  9. [15]

    Positive scalar curvature metrics and aspherical summands

    Shuli Chen, Jianchun Chu, and Jintian Zhu. Positive scalar curvature metrics and aspherical summands. 2024. arXiv: 2312.04698

    2024

  10. [16]

    3-Manifolds with positive scalar curvature and bounded geometry

    Otis Chodosh, Yi Lai, and Kai Xu. 3-Manifolds with positive scalar curvature and bounded geometry. 2025. arXiv: 2502.09727

    2025 arXiv

  11. [17]

    Generalized soap bubbles and the topology of manifolds with positive scalar curvature

    Otis Chodosh and Chao Li. “Generalized soap bubbles and the topology of manifolds with positive scalar curvature”. In: Ann. of Math. (2) 199.2 (2024), pp. 707–740

    2024

  12. [18]

    Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

    Otis Chodosh, Chao Li, and Yevgeny Liokumovich. “Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions”. In: Geom. Topol. 27.4 (2023), pp. 1635–1655

    2023

  13. [20]

    Positivity of curvature on manifolds with boundary

    Tsz-Kiu Aaron Chow. “Positivity of curvature on manifolds with boundary”. In: Int. Math. Res. Not. IMRN 15 (2022), pp. 11401–11426

    2022

  14. [21]

    K¨ urzeste Wege und Totalkr¨ ummung auf Fl¨ achen

    Stefan Cohn-Vossen. “K¨ urzeste Wege und Totalkr¨ ummung auf Fl¨ achen”. In:Compo- sitio Math. 2 (1935), pp. 69–133

    1935

  15. [22]

    Three-manifolds with bounded curvature and uniformly positive scalar curvature

    Conghan Dong. “Three-manifolds with bounded curvature and uniformly positive scalar curvature”. In: J. Geom. Anal. 33.6 (2023), Paper No. 169, 11

    2023

  16. [23]

    Compactness of the space of embedded min- imal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

    Ailana Fraser and Martin Man-chun Li. “Compactness of the space of embedded min- imal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary”. In: J. Differential Geom. 96.2 (2014), pp. 183–200

    2014

  17. [24]

    The topology of four-dimensional manifolds

    Michael Freedman. “The topology of four-dimensional manifolds”. In: J. Differential Geometry 17.3 (1982), pp. 357–453

    1982

  18. [25]

    ¨Uber die Enden diskreter R¨ aume und Gruppen

    Hans Freudenthal. “ ¨Uber die Enden diskreter R¨ aume und Gruppen”. In: Comment. Math. Helv. 17 (1945), pp. 1–38

    1945

  19. [26]

    Foliations and the topology of 3-manifolds. III

    David Gabai. “Foliations and the topology of 3-manifolds. III”. In: J. Differential Geom. 26.3 (1987), pp. 479–536

    1987

  20. [27]

    On complete open manifolds of positive curva- ture

    Detlef Gromoll and Wolfgang Meyer. “On complete open manifolds of positive curva- ture”. In: Ann. of Math. (2) 90 (1969), pp. 75–90

    1969

  21. [28]

    Partial differential relations

    Mikhael Gromov. Partial differential relations . Vol. 9. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer- Verlag, Berlin, 1986, pp. x+363

    1986

  22. [29]

    The classification of simply connected manifolds of positive scalar curvature

    Mikhael Gromov and H. Blaine Lawson Jr. “The classification of simply connected manifolds of positive scalar curvature”. In: Ann. of Math. (2) 111.3 (1980), pp. 423– 434

    1980

  23. [30]

    Four lectures on scalar curvature

    Misha Gromov. “Four lectures on scalar curvature”. In: Perspectives in scalar curva- ture. Vol. 1 . World Sci. Publ., Hackensack, NJ, 2023, pp. 1–514

    2023

  24. [31]

    Four-manifolds with positive isotropic curvature

    Richard Hamilton. “Four-manifolds with positive isotropic curvature”. In: Comm. Anal. Geom. 5.1 (1997), pp. 1–92

    1997

  25. [32]

    Three-manifolds with positive Ricci curvature

    Richard Hamilton. “Three-manifolds with positive Ricci curvature”. In: J. Differential Geometry 17.2 (1982), pp. 255–306

    1982

  26. [33]

    3 -Manifolds

    John Hempel. 3 -Manifolds. Vol. No. 86. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1976, pp. xii+195

    1976

  27. [34]

    Enden offener R¨ aume und unendliche diskontinuierliche Gruppen

    Heinz Hopf. “Enden offener R¨ aume und unendliche diskontinuierliche Gruppen”. In: Comment. Math. Helv. 16 (1944), pp. 81–100. 20 REFERENCES

    1944

  28. [35]

    Classification of compact manifolds with positive isotropic curvature

    Hong Huang. Classification of compact manifolds with positive isotropic curvature

  29. [36]

    Compact manifolds of dimension n ≥ 12 with positive isotropic curva- ture

    Hong Huang. Compact manifolds of dimension n ≥ 12 with positive isotropic curva- ture. 2024. arXiv: 1909.12265

    2024 arXiv

  30. [37]

    On subharmonic functions and differential geometry in the large

    Alfred Huber. “On subharmonic functions and differential geometry in the large”. In: Comment. Math. Helv. 32 (1957), pp. 13–72

    1957

  31. [38]

    Smooth homology spheres and their fundamental groups

    Michel Kervaire. “Smooth homology spheres and their fundamental groups”. In: Trans. Amer. Math. Soc. 144 (1969), pp. 67–72

    1969

  32. [39]

    Embedding and surrounding with positive mean curvature

    H. Blaine Lawson Jr. and Marie-Louise Michelsohn. “Embedding and surrounding with positive mean curvature”. In: Invent. Math. 77.3 (1984), pp. 399–419

    1984

  33. [40]

    Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature

    William Meeks III, Leon Simon, and Shing Tung Yau. “Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature”. In: Ann. of Math. (2) 116.3 (1982), pp. 621–659

    1982

  34. [41]

    Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes

    Mario Micallef and John Douglas Moore. “Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes”. In: Ann. of Math. (2) 127.1 (1988), pp. 199–227

    1988

  35. [42]

    A note on curvature and fundamental group

    John Milnor. “A note on curvature and fundamental group”. In: J. Differential Ge- ometry 2 (1968), pp. 1–7

    1968

  36. [43]

    Lectures on the h-cobordism theorem

    John Milnor. Lectures on the h-cobordism theorem. Princeton University Press, Prince- ton, NJ, 2025, p. 113

    2025

  37. [44]

    Absolutely graded Floer homologies and intersec- tion forms for four-manifolds with boundary

    Peter Ozsv´ ath and Zolt´ an Szab´ o. “Absolutely graded Floer homologies and intersec- tion forms for four-manifolds with boundary”. In: Adv. Math. 173.2 (2003), pp. 179– 261

    2003

  38. [45]

    Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers

    Grigori Perelman. “Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers”. In: Comparison geometry (Berkeley, CA, 1993– 94). Vol. 30. Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 1997, pp. 157–163

    1993

  39. [46]

    Proof of the soul conjecture of Cheeger and Gromoll

    Grigori Perelman. “Proof of the soul conjecture of Cheeger and Gromoll”. In: J. Dif- ferential Geom. 40.1 (1994), pp. 209–212

    1994

  40. [47]

    Ricci flow with surgery on three-manifolds

    Grigori Perelman. Ricci flow with surgery on three-manifolds . 2003. arXiv: math / 0303109

    2003

  41. [48]

    Some results on nonnegatively curved manifolds

    Walter A. Poor Jr. “Some results on nonnegatively curved manifolds”. In: J. Differ- ential Geometry 9 (1974), pp. 583–600

    1974

  42. [49]

    Scalar and mean curvature comparison via µ-bubbles

    Daniel R¨ ade. “Scalar and mean curvature comparison via µ-bubbles”. In: Calc. Var. Partial Differential Equations 62.7 (2023), Paper No. 187, 39

    2023

  43. [50]

    Manifolds of positive scalar curvature: a progress report

    Jonathan Rosenberg. “Manifolds of positive scalar curvature: a progress report”. In: Surveys in differential geometry. Vol. XI . Vol. 11. Surv. Differ. Geom. Int. Press, Somerville, MA, 2007, pp. 259–294

    2007

  44. [51]

    Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature

    R. Schoen and Shing Tung Yau. “Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature”. In: Ann. of Math. (2) 110.1 (1979), pp. 127–142

    1979

  45. [52]

    On the structure of manifolds with positive scalar curvature

    Richard Schoen and Shing-Tung Yau. “On the structure of manifolds with positive scalar curvature”. In: Manuscripta Math. 28.1-3 (1979), pp. 159–183

    1979

  46. [53]

    p-convex Riemannian manifolds

    Ji-Ping Sha. “ p-convex Riemannian manifolds”. In:Invent. Math. 83.3 (1986), pp. 437– 447

    1986

  47. [54]

    Handlebodies and p-convexity

    Ji-Ping Sha. “Handlebodies and p-convexity”. In: J. Differential Geom. 25.3 (1987), pp. 353–361. REFERENCES 21

    1987

  48. [55]

    Examples of manifolds of positive Ricci curvature

    Ji-Ping Sha and DaGang Yang. “Examples of manifolds of positive Ricci curvature”. In: J. Differential Geom. 29.1 (1989), pp. 95–103

    1989

  49. [56]

    Positive Ricci curvature on the connected sums of Sn × Sm

    Ji-Ping Sha and DaGang Yang. “Positive Ricci curvature on the connected sums of Sn × Sm”. In: J. Differential Geom. 33.1 (1991), pp. 127–137

    1991

  50. [57]

    Homology spheres which are covered by spheres

    Denis Sjerve. “Homology spheres which are covered by spheres”. In: J. London Math. Soc. (2) 6 (1973), pp. 333–336

    1973

  51. [58]

    Generalized Poincar´ e’s conjecture in dimensions greater than four

    Stephen Smale. “Generalized Poincar´ e’s conjecture in dimensions greater than four”. In: Ann. of Math. (2) 74 (1961), pp. 391–406.issn: 0003-486X. doi: 10.2307/1970239. url: https://doi.org/10.2307/1970239

    1961 doi

  52. [59]

    On the structure of manifolds

    Stephen Smale. “On the structure of manifolds”. In: Amer. J. Math. 84 (1962), pp. 387–399

    1962

  53. [60]

    The piecewise-linear structure of Euclidean space

    John Stallings. “The piecewise-linear structure of Euclidean space”. In: Proc. Cam- bridge Philos. Soc. 58 (1962), pp. 481–488

    1962

  54. [61]

    A volume invariant of coverings

    Albert ˇSvarc. “A volume invariant of coverings”. In: Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), pp. 32–34

    1955

  55. [63]

    Poincar´ e complexes. I

    C. T. C. Wall. “Poincar´ e complexes. I”. In: Ann. of Math. (2) 86 (1967), pp. 213–245

    1967

  56. [64]

    Boundary convexity on manifolds with nonnegative Ricci curva- ture

    Hui-Hsien Wang. “Boundary convexity on manifolds with nonnegative Ricci curva- ture”. In: Pacific J. Math. 191.2 (1999), pp. 393–398

    1999

  57. [65]

    Contractible 3-manifolds and positive scalar curvature (I)

    Jian Wang. “Contractible 3-manifolds and positive scalar curvature (I)”. In: J. Dif- ferential Geom. 127.3 (2024), pp. 1267–1304

    2024

  58. [66]

    Contractible 3-manifolds and positive scalar curvature (II)

    Jian Wang. “Contractible 3-manifolds and positive scalar curvature (II)”. In: J. Eur. Math. Soc. (JEMS) 26.2 (2024), pp. 537–572

    2024

  59. [67]

    Topology of 3-manifolds with uniformly positive scalar curvature

    Jian Wang. Topology of 3-manifolds with uniformly positive scalar curvature . 2023. arXiv: 2212.14383

    2023 arXiv

  60. [68]

    Capillary surfaces in manifolds with nonnegative scalar curvature and strictly mean convex boundary

    Yujie Wu. “Capillary surfaces in manifolds with nonnegative scalar curvature and strictly mean convex boundary”. In: Int. Math. Res. Not. IMRN 9 (2025), Paper No. rnaf106, 14

    2025

  61. [69]

    Problem section

    Shing-Tung Yau. “Problem section”. In: Seminar on Differential Geometry . Vol. 102. Annals of Mathematics Studies. Princeton, N.J.: Princeton University Press, 1982, pp. 669–706

    1982

  62. [70]

    Width estimate and doubly warped product

    Jintian Zhu. “Width estimate and doubly warped product”. In: Trans. Amer. Math. Soc. 374.2 (2021), pp. 1497–1511. Paul Sweeney Jr., Universit `a di Trento, Dipartimento di Matematica, via Sommarive 14, 38123 Povo di Trento, Italy Email address: paul.sweeneyjr@unitn.it

    2021

Pith tools

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