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Band Width Estimates and Rigidity of Manifolds with Negative Curvature

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Pith's one-line read Three-dimensional manifolds with Ricci curvature bounded below by negative constants admit optimal Lipschitz lower bounds for proper smooth functions.

desk verdict Cruz gives new width estimates for 3-manifolds with negative Ricci bounds via μ-bubbles, extending to higher-genus boundaries plus a rigidity result for noncompact cases with controlled H2. read the letter →

arxiv 2606.25132 v2 pith:5FLPCIFU submitted 2026-06-23 math.DG

Tiarlos Cruz This is my paper
classification math.DG
keywords RiemannianmanifoldsRiccicurvaturewidthestimatesLipschitzboundsrigiditythree-manifoldsnegativebands
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 establishes optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants. These bounds produce new width estimates for Riemannian bands. The approach handles higher-genus boundary components and supplies rigidity statements for equality cases. It also yields a sharp lower bound on boundary area for certain complete noncompact three-manifolds with scalar curvature at least -6 and no spherical or toroidal classes in second homology, with equality only for the infinite hyperbolic band.

What carries the argument

Gromov's μ-bubble method, which produces the Lipschitz lower bounds and width estimates while handling higher-genus boundary components under the Ricci curvature lower bound.

What would settle it

A three-dimensional Riemannian manifold with Ricci curvature bounded below by a negative constant that admits a proper smooth function whose Lipschitz constant falls below the optimal lower bound claimed in the theorem.

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Extended reading notes

Core claim

We establish optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants, yielding a new family of width estimates for Riemannian bands using Gromov's μ-bubble method, together with rigidity statements characterizing the equality case. One of the novelties of our approach lies in its ability to handle higher-genus boundary components, revealing a precise interplay between the width, the area of boundary surfaces, and the underlying topology. Finally, for a complete noncompact three-manifold M with bounded geometry and scalar curvature R_g ≥ -6, whose H_2(M,ℤ) contains no spherical or toroidal cl

Load-bearing premise

The μ-bubble method can be applied to produce the stated Lipschitz and width bounds while handling higher-genus boundary components under the given Ricci lower bound.

Editorial extensions

If this is right

  • New family of width estimates for Riemannian bands in three-manifolds satisfying the Ricci lower bound.
  • Rigidity statements that characterize the manifolds achieving equality in the Lipschitz and width bounds.
  • Width estimates that incorporate the genus of the boundary components and relate width to boundary area and topology.
  • Sharp lower bound on boundary area for complete noncompact three-manifolds with bounded geometry, scalar curvature at least -6, and no spherical or toroidal homology classes.
  • Equality case in the area bound is an infinite hyperbolic band.

Reading between the lines

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

  • The interplay between width, boundary area, and topology could be examined in manifolds satisfying different curvature lower bounds.
  • The homology restriction in the noncompact result limits the manifolds to which the area bound applies.
  • The sharpness of the bounds could be tested by direct computation on explicit examples such as quotients of hyperbolic three-space.
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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 / 2 minor

Summary. The manuscript establishes optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants. These bounds are applied via Gromov's μ-bubble method to obtain a new family of width estimates for Riemannian bands, including those with higher-genus boundary components, together with rigidity statements characterizing equality cases. For complete noncompact 3-manifolds with bounded geometry, scalar curvature R_g ≥ -6, and H_2(M, ℤ) containing no spherical or toroidal classes, a sharp lower bound on boundary area is proved, with equality implying the manifold is isometric to an infinite hyperbolic band.

Significance. If the central claims hold, the work provides a meaningful extension of the μ-bubble technique to the negative-curvature setting and to higher-genus boundaries, yielding explicit quantitative estimates that relate width, boundary area, and topology. The rigidity results and the sharp area bound under the stated topological hypothesis are potentially useful contributions to the study of 3-manifolds with curvature bounds. The absence of free parameters or ad-hoc fitting in the stated results is a strength.

minor comments (2)
  1. [Introduction] The abstract and introduction refer to 'optimal' Lipschitz bounds and 'sharp' area bounds; a brief remark in §1 or §2 clarifying the sense in which optimality is achieved (e.g., equality on the model hyperbolic band) would help readers.
  2. [Theorem on noncompact manifolds] In the statement of the noncompact area bound, the precise meaning of 'bounded geometry' (e.g., injectivity radius and curvature bounds) should be recorded explicitly, as it is used to control the μ-bubble construction at infinity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation applies external μ-bubble method to new estimates

full rationale

The paper's claims rest on applying Gromov's established μ-bubble method (explicitly cited as external) to obtain Lipschitz lower bounds and width estimates for bands under a negative Ricci lower bound, including handling of higher-genus boundaries. The abstract and described novelties present these as new applications yielding rigidity to hyperbolic bands when equality holds, with topological conditions on H_2 serving as hypotheses rather than derived outputs. No equations reduce by construction to fitted parameters, no self-citation chains justify the central premise, and no ansatz or renaming is smuggled in. The derivation chain is therefore self-contained against the external method and curvature assumptions.

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

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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Cite this review

Pith. "Pith review of Band Width Estimates and Rigidity of Manifolds with Negative Curvature." pith.science (2026). https://pith.science/paper/5FLPCIFU

@misc{pith2026260625132,
  author       = {Pith},
  title        = {Pith review of: Band Width Estimates and Rigidity of Manifolds with Negative Curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FLPCIFU}},
  note         = {Machine review of arXiv:2606.25132}
}
abstract

We establish optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants, yielding a new family of width estimates for Riemannian bands using Gromov's $\mu$-bubble method, together with rigidity statements characterizing the equality case. One of the novelties of our approach lies in its ability to handle higher-genus boundary components, revealing a precise interplay between the width, the area of boundary surfaces, and the underlying topology. Finally, for a complete noncompact three-manifold $M$ with bounded geometry and scalar curvature $R_g\ge -6$, whose $H_2(M,\mathbb{Z})$ contains no spherical or toroidal classes, we prove a sharp lower bound for the boundary area. In the equality case, the manifold is shown to be isometric to an infinite hyperbolic band.

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Andersson, M.-L

    L. Andersson, M.-L. Cai, G. Galloway,Rigidity and positivity of mass for asymptoti- cally hyperbolic manifolds.Ann. Henri Poincar´ e 9(1), 1–33 (2008)

    work page 2008

  2. [2]

    A sharp spectral splitting theorem

    G. Antonelli, M. Pozzetta, K. Xu,A sharp spectral splitting theorem.Available at arXiv:2412.12707

    work page Pith review arXiv

  3. [3]

    Aubin,Metriques riemanniennes et courbure,J

    T. Aubin,Metriques riemanniennes et courbure,J. Diff. Geom. 4(1970), 383-424

    work page 1970

  4. [4]

    Bland and M

    J. Bland and M. Kalka,Negative scalar curvature metrics on noncompact manifolds, Trans. A.M.S., No. 2 (1989), 433-446. 18

    work page 1989

  5. [5]

    H. Bray, S. Brendle, M. Eichmair, and A. Neves,Area-minimizing projective planes in 3-manifolds,Commun. Pure Appl. Math. 63 (2010), no. 9, 1237–1247 (English)

    work page 2010

  6. [6]

    Carlotto, O

    A. Carlotto, O. Chodosh, and M. Eichmair.Effective versions of the positive mass theorem.Invent. Math., 206(3):975–1016, 2016

    work page 2016

  7. [7]

    Cecchini,A long neck principle for Riemannian spin manifolds with positive scalar curvature,Geom

    S. Cecchini,A long neck principle for Riemannian spin manifolds with positive scalar curvature,Geom. Funct. Anal. 30 (2020), 1183-1223

    work page 2020

  8. [8]

    Cecchini, R

    S. Cecchini, R. Zeidler,Scalar and mean curvature comparison via the Dirac operator, arXiv:2103.06833v1, March 2021

    work page arXiv 2021

Show all 43 references
  1. [9]

    Cecchini, D

    S. Cecchini, D. R¨ ade, and R. Zeidler,Nonnegative scalar curvature on manifolds with at least two ends,J. Topol. 16 (2023), no. 3, 855–876 (English)

    2023

  2. [10]

    Chai, and Y

    X. Chai, and Y. Sun.Band width estimates with lower spectral curvature bounds. arXiv:2504.10142, April 2025

    2025 arXiv

  3. [11]

    Cheeger and D

    J. Cheeger and D. Gromoll,The splitting theorem for manifolds of nonnegative Ricci curvature,J. Differential Geometry 6 (1971/72), 119–128. MR 303460

    1971

  4. [12]

    Chodosh, M

    O. Chodosh, M. Eichmair, and V. Moraru.A splitting theorem for scalar curvature. Comm. Pure Appl. Math., 72(6):1231–1242, 2019

    2019

  5. [13]

    Chodosh and C

    O. Chodosh and C. Li.Stable anisotropic minimal hypersurfaces inR 4,Forum Math. Pi 11 (2023), 22 (English), Id/No e3

    2023

  6. [14]

    Chodosh and C

    O. Chodosh and C. Li.Generalized soap bubbles and the topology of manifolds with positive scalar curvature. Ann. of Math. (2) 199.2 (2024), pp. 707–740

    2024

  7. [15]

    C. B. Croke, and B. Kleiner,A warped product splitting theorem.Duke Math. J. 67, 3 (1992), 571–574

    1992

  8. [16]

    Cruz and F

    T. Cruz and F. Vit´ orio.Prescribing the curvature of Riemannian manifolds with boundary.Calculus of Variations and Partial Differential Equations, 58(4):124, 2019

    2019

  9. [17]

    Eichmair, G

    M. Eichmair, G. Galloway, A. Mendes,Initial data rigidity results.Commun. Math. Phys. 386(1), 253–268 (2021)

    2021

  10. [18]

    Frensel,Stable complete surfaces with constant mean curvature.Boletim da Sociedade Brasileira de Matem´ atica, 27, (1996) 129-144

    K.R. Frensel,Stable complete surfaces with constant mean curvature.Boletim da Sociedade Brasileira de Matem´ atica, 27, (1996) 129-144

    1996

  11. [19]

    Gao and S.T

    Z. Gao and S.T. Yau,The existence of negatively Ricci curved metrics on three man- ifolds,Invent. Math. 85(1986), 637-652

    1986

  12. [20]

    Giusti.Minimal surfaces and functions of bounded variation, volume 80 of Mono- graphs in Mathematics.Birkh¨ auser Verlag, Basel, 1984

    E. Giusti.Minimal surfaces and functions of bounded variation, volume 80 of Mono- graphs in Mathematics.Birkh¨ auser Verlag, Basel, 1984

    1984

  13. [21]

    Gromov,Positive curvature, macroscopic dimension, spectral gaps and higher signatures,Functional analysis on the eve of the 21st century

    M. Gromov,Positive curvature, macroscopic dimension, spectral gaps and higher signatures,Functional analysis on the eve of the 21st century. Volume II. In honor of the eightieth birthday of I. M. Gelfand. Proceedings of a conference, held at Rutgers University, New Brunswick, ...

    1993

  14. [22]

    Gromov.Four lectures on scalar curvature.World Scientific Publishing Co.Pte

    M. Gromov.Four lectures on scalar curvature.World Scientific Publishing Co.Pte. Ltd., Hackensack, NJ, 2023, 1–514. ISBN: 978-981-124-998-3; 978-981-124-935-8; 978- 981-124-936-5

    2023

  15. [23]

    Gromov,Metric inequalities with scalar curvature,Geom

    M. Gromov,Metric inequalities with scalar curvature,Geom. Funct. Anal. 28 (2018), 645-726

    2018

  16. [24]

    Gromov.Scalar curvature of manifolds with boundaries: natural questions and artificial constructions.In: arXiv preprint arXiv:1811.04311 (2018)

    M. Gromov.Scalar curvature of manifolds with boundaries: natural questions and artificial constructions.In: arXiv preprint arXiv:1811.04311 (2018)

    2018 arXiv

  17. [25]

    Hirsch, D

    S. Hirsch, D. Kazaras, M. Khuri, Y. Zhang,Rigid comparison geometry for Riemannian bands and open incomplete manifolds,Math. Ann. (2024), doi.org/10.1007/s00208-024-02973-y

    2024 doi

  18. [26]

    T. Hao, Y. Hu, P. Liu, and Y. Shi,Rigidity and Non-Rigidity ofH n/Zn−2 with Scalar Curvature Bounded from Below.SIGMA 19 (2023), 083, 28 pages

    2023

  19. [27]

    He and J

    S. He and J. ZhuA note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds.Proc. Amer. Math. Soc. 153 (2025), no. 2, 829-840

    2025

  20. [28]

    Kazdan and F

    J. Kazdan and F. W. Warner,Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures,Ann. of Math. (2) 101 (1975), 317–331

    1975

  21. [29]

    Lesourd, R

    M. Lesourd, R. Unger, and S-T Yau,The positive mass theorem with arbitrary ends, J. Differ. Geom. 128 (2024), no. 1, 257–293 (English)

    2024

  22. [30]

    Lohkamp,Metrics of negative Ricci curvature, Annals of Mathematics,140 (1994), 655-683

    J. Lohkamp,Metrics of negative Ricci curvature, Annals of Mathematics,140 (1994), 655-683. 19

    1994

  23. [31]

    Liu,3-manifolds with nonnegative Ricci curvature

    G. Liu,3-manifolds with nonnegative Ricci curvature. Invent. Math. 193.2 (2013), pp. 367- 375

    2013

  24. [32]

    F. C. Marques, Andr´ e Neves.Rigidity of min-max minimal spheres in three-manifolds. Duke Math. J., 161(14):2725–2752, 2012

    2012

  25. [33]

    Mazet,Stable minimal hypersurfaces inR 6,Preprint, arXiv:2405.14676 [math.DG], 2024

    L. Mazet,Stable minimal hypersurfaces inR 6,Preprint, arXiv:2405.14676 [math.DG], 2024

    2024

  26. [34]

    Nunes.Rigidity of area-minimizing hyperbolic surfaces in three-manifolds.J

    I. Nunes.Rigidity of area-minimizing hyperbolic surfaces in three-manifolds.J. Geom. Anal., 23(3):1290–1302, 2013

    2013

  27. [35]

    R¨ ade.Scalar and mean curvature comparison viaµ-bubbles.Calculus of Variations and Partial Differential Equations, 62(7):187, 2023

    D. R¨ ade.Scalar and mean curvature comparison viaµ-bubbles.Calculus of Variations and Partial Differential Equations, 62(7):187, 2023

    2023

  28. [36]

    Schoen and S-T

    R. Schoen and S-T. Yau,On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700

    1979

  29. [37]

    Yau,Some function-theoretic properties of complete Riemannian manifold and their applications to geometry,Indiana Univ

    S-T. Yau,Some function-theoretic properties of complete Riemannian manifold and their applications to geometry,Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452

    1976

  30. [38]

    Wang,Topology of 3-manifolds with uniformly positive scalar curvature,Preprint, arXiv:2212.14383 [math.DG], 2022

    J. Wang,Topology of 3-manifolds with uniformly positive scalar curvature,Preprint, arXiv:2212.14383 [math.DG], 2022

    2022

  31. [39]

    Zeidler,Band width estimates via the Dirac operator.Journal of Differential Ge- ometry, v

    R. Zeidler,Band width estimates via the Dirac operator.Journal of Differential Ge- ometry, v. 122, n. 1, p. 155-183, 2022

    2022

  32. [40]

    Zhou and J

    X. Zhou and J. Zhu.Existence of hypersurfaces with prescribed mean cur- vature I—generic min-max.Camb. J. Math. 8 (2020), no.2, 311–362

    2020

  33. [41]

    Zhu.Width estimate and doubly warped product,Trans

    J. Zhu.Width estimate and doubly warped product,Trans. Amer. Math. Soc. 374 (2021), no. 2, 1497–1511. MR 4196400

    2021

  34. [42]

    Zhu.Rigidity results for complete manifolds with nonnegative scalar curvature.J

    J. Zhu.Rigidity results for complete manifolds with nonnegative scalar curvature.J. Differential Geom. 125.3 (2023), pp. 623–644

    2023

  35. [43]

    Zhu.Calabi-Yau type theorem for complete manifolds with nonnegative scalar cur- vature.available at https: // arxiv

    J. Zhu.Calabi-Yau type theorem for complete manifolds with nonnegative scalar cur- vature.available at https: // arxiv. org/ abs/ 2402. 15118 , 2024. Institute of Mathematics, Federal University of Alagoas, 57072-970, Macei´o-AL, Brazil Email address:cicero.cruz@im.ufal.br 20

    2024

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