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arxiv: 2604.21002 · v2 · submitted 2026-04-22 · 🧮 math.DG

Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

Samuel Belo This is my paper

Pith reviewed 2026-05-09 22:51 UTC · model grok-4.3

classification 🧮 math.DG
keywords m-quasi-Einstein manifoldsdiameter estimatesHitchin-Thorpe inequalitypotential functionoscillationfour-dimensional manifoldsRicci solitonsvolume estimates
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The pith

Lower bounds on the diameter of compact m-quasi-Einstein manifolds are derived from the oscillation of the potential function, providing conditions for the Hitchin-Thorpe inequality in four dimensions.

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

The paper derives lower bounds for the diameter of compact m-quasi-Einstein manifolds based on the oscillation of the potential function. This relation provides geometric control over the size of the manifold. Applying this in four dimensions produces diameter conditions that guarantee the manifolds satisfy the Hitchin-Thorpe inequality. The work extends estimates from smooth metric measure spaces and recovers known results for Ricci solitons as a limit case. It also includes a volume estimate tied to the oscillation.

Core claim

We study compact m-quasi-Einstein manifolds and derive geometric estimates relating the oscillation of the potential function to the diameter of the manifold. We obtain lower bounds for the diameter in terms of the oscillation of the potential function. As an application in dimension four, we derive diameter conditions ensuring that compact m-quasi-Einstein manifolds satisfy the Hitchin--Thorpe inequality. Our results extend diameter estimates in smooth metric measure spaces and are consistent with known bounds in the limiting case corresponding to Ricci solitons. Finally, we provide a volume estimate involving the oscillation.

What carries the argument

The m-quasi-Einstein equation and associated integral estimates that link the oscillation of the potential function to lower bounds on the manifold's diameter.

If this is right

  • Lower bounds for the diameter are obtained in terms of the oscillation of the potential function.
  • Diameter conditions in four dimensions ensure that the manifolds satisfy the Hitchin-Thorpe inequality.
  • The estimates extend those for smooth metric measure spaces.
  • Consistency with bounds for Ricci solitons is established.
  • A volume estimate involving the oscillation is derived.

Where Pith is reading between the lines

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

  • The derived estimates could be tested for sharpness using known explicit examples of quasi-Einstein manifolds.
  • Similar techniques might yield analogous bounds in higher dimensions for other curvature inequalities.
  • Connections to the topology of the manifold beyond the four-dimensional case may be explored through these diameter controls.

Load-bearing premise

The manifold is assumed to be smooth, compact, and m-quasi-Einstein with a well-defined smooth potential function that controls the diameter through its oscillation.

What would settle it

Construction of a compact four-dimensional m-quasi-Einstein manifold where the diameter is less than the lower bound predicted by the oscillation of the potential, or which violates the Hitchin-Thorpe inequality while meeting the diameter condition.

read the original abstract

We study compact $m$-quasi-Einstein manifolds and derive geometric estimates relating the oscillation of the potential function to the diameter of the manifold. We obtain lower bounds for the diameter in terms of the oscillation of the potential function. As an application in dimension four, we derive diameter conditions ensuring that compact $m$-quasi-Einstein manifolds satisfy the Hitchin--Thorpe inequality. Our results extend diameter estimates in smooth metric measure spaces and are consistent with known bounds in the limiting case corresponding to Ricci solitons. Finally, we provide a volume estimate involving the oscillation.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript studies compact m-quasi-Einstein manifolds and derives lower bounds on the diameter in terms of the oscillation of the potential function. As an application in dimension four, it obtains diameter conditions guaranteeing that such manifolds satisfy the Hitchin-Thorpe inequality. The work also supplies a volume estimate involving the oscillation and asserts consistency with known bounds for Ricci solitons as well as extensions of diameter estimates from smooth metric measure spaces.

Significance. If the derivations hold, the results furnish concrete geometric constraints linking the potential oscillation to diameter and topology in the quasi-Einstein setting. The four-dimensional application to the Hitchin-Thorpe inequality is a natural and potentially useful extension, while the consistency with the Ricci-soliton limit and the volume estimate add incremental value to the literature on smooth metric measure spaces.

minor comments (3)
  1. The introduction would benefit from an explicit recall of the m-quasi-Einstein equation (including the precise roles of m, λ, and the potential f) to improve accessibility for readers outside the immediate subfield.
  2. Notation for the oscillation of the potential (e.g., Osc(f) or equivalent) should be introduced once and used uniformly in all statements of the main theorems and corollaries.
  3. The volume estimate involving the oscillation is mentioned in the abstract but its precise statement and proof location should be highlighted in the introduction or a dedicated subsection for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on diameter estimates for compact m-quasi-Einstein manifolds and the application to the Hitchin-Thorpe inequality in dimension four. The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no points requiring detailed rebuttal or revision at this stage. We will incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation starts from the standard m-quasi-Einstein equation on a compact manifold and obtains diameter lower bounds by integrating against the potential function and applying gradient estimates. These steps are independent of the final statements and do not reduce to the inputs by construction. The four-dimensional application simply combines the resulting diameter condition with the known topological form of the Hitchin-Thorpe inequality. No self-citations are load-bearing, no parameters are fitted and then relabeled as predictions, and no ansatz is smuggled via prior work by the same author. The results are consistent with the Ricci-soliton limit but are not defined by it. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the work relies on standard background from Riemannian geometry and smooth metric measure spaces.

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    and ´Emery, M.,Diffusions hypercontractives, S´ eminaire de Probabilit´ es XIX 1983/84, Lecture Notes in Math.1123, Springer, Berlin, (1985), 177–206

    Bakry, D. and ´Emery, M.,Diffusions hypercontractives, S´ eminaire de Probabilit´ es XIX 1983/84, Lecture Notes in Math.1123, Springer, Berlin, (1985), 177–206

    work page 1983

  2. [2]

    and Ribeiro Jr., E.,Integral formulae on quasi-Einstein manifolds and applications, Glasg

    Barros, A. and Ribeiro Jr., E.,Integral formulae on quasi-Einstein manifolds and applications, Glasg. Math. J.54(2012), 213–223

    work page 2012

  3. [3]

    Barros, A., Ribeiro Jr., E., and Silva, J.,Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds, Diff. Geom. Appl.35(2014), 60–73

    work page 2014

  4. [4]

    J., Jizany, A., and Murphy, T.,Conformally K¨ ahler geometry and quasi-Einstein metrics, M¨ unster J

    Batat, W., Hall, S. J., Jizany, A., and Murphy, T.,Conformally K¨ ahler geometry and quasi-Einstein metrics, M¨ unster J. Math.8(2015), 211–228

    work page 2015

  5. [5]

    L.,Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008

    Besse, A. L.,Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008

    work page 2008

  6. [6]

    Math.134(1998), 145–176

    B¨ ohm, C.,Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math.134(1998), 145–176

    work page 1998

  7. [7]

    B¨ ohm, C.,Non-compact cohomogeneity one Einstein manifolds, Bull. Soc. Math. France127(1999), 135–177

    work page 1999

  8. [8]

    Brasil, A., Costa, E., and Ribeiro Jr., E.,Hitchin–Thorpe inequality and Kaehler metrics for compact almost Ricci soliton, Ann. Mat. Pura Appl. (4)193(2014), no. 6, 1851–1860

    work page 2014

  9. [9]

    Cao, H.-D.,Recent progress on Ricci solitons, Adv. Lect. Math.11(2010), 1–38

    work page 2010

  10. [10]

    Case, J., Shu, Y.-J., and Wei, G.,Rigidity of quasi-Einstein metrics, Differ. Geom. Appl.29(2011), 93–100

    work page 2011

  11. [11]

    Catino, G.,A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds, Diff. Geom. Appl. 30(2012), 660–664

    work page 2012

  12. [12]

    Reine Angew

    Catino, G., Mantegazza, C., Mazzieri, L., and Rimoldi, M.,Locally conformally flat quasi-Einstein manifolds, J. Reine Angew. Math. (Crelle’s Journal)675(2013), 181–189

    work page 2013

  13. [13]

    Cheng, X., Ribeiro Jr., E., and Zhou, D.,Volume growth estimates for Ricci solitons and quasi-Einstein manifolds, J. Geom. Anal.32(2022), 62

    work page 2022

  14. [14]

    Cheng, X., Ribeiro Jr., E., and Zhou, D.,On Euler characteristic and Hitchin–Thorpe inequality for compact gradient Ricci solitons, Proc. Amer. Math. Soc.10(2023), 33–45

    work page 2023

  15. [15]

    Colding and W.P

    Colding, T. and Minicozzi, W.,Optimal growth bounds for eigenfunctions, arXiv preprint arXiv:2109.04998 (2021)

    work page arXiv 2021

  16. [16]

    Costa, J., Ribeiro Jr., E., and Santos, M.,On quasi-Einstein manifolds with constant scalar curvature, arXiv preprint arXiv:2505.18834 (2025)

    work page arXiv 2025

  17. [17]

    Deng, Y.,Diameter estimate for a class of compact generalized quasi-Einstein manifolds, Turk. J. Math.45(2021), 2331–2340

    work page 2021

  18. [18]

    Deng, Y.,Diameter estimate for generalizedm-quasi-Einstein manifolds, Ital. J. Pure Appl. Math.52 (2024), 9–17

    work page 2024

  19. [19]

    and Gadelha, T.,Compact quasi-Einstein manifolds with boundary, Math

    Di´ ogenes, R. and Gadelha, T.,Compact quasi-Einstein manifolds with boundary, Math. Nachr.295 (2022), 1690–1708

    work page 2022

  20. [20]

    Di´ ogenes, R., Gadelha, T., and Ribeiro Jr., E.,Remarks on quasi-Einstein manifolds with boundary, Proc. Amer. Math. Soc.150(2022), 351–363

    work page 2022

  21. [21]

    Di´ ogenes, R., Gon¸ calves, J., and Ribeiro Jr., E.,Geometric inequalities for quasi-Einstein manifolds, Ann. Mat. Pura Appl. (2025)

    work page 2025

  22. [22]

    and Garc´ ıa-R´ ıo, E.,Diameter bounds and Hitchin–Thorpe inequalities for compact Ricci solitons, Quart

    Fern´ andez-L´ opez, M. and Garc´ ıa-R´ ıo, E.,Diameter bounds and Hitchin–Thorpe inequalities for compact Ricci solitons, Quart. J. Math.61(2010), 319–327

    work page 2010

  23. [23]

    J.,Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic, Indiana Univ

    Gursky, M. J.,Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic, Indiana Univ. Math. J.43(1994), 747–774

    work page 1994

  24. [24]

    II (Cambridge, MA, 1993), Int

    Hamilton, R.,The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, (1995), 7–136. 16 SAMUEL BELO

    work page 1993

  25. [25]

    He, C., Petersen, P., and Wylie, W.,On the classification of warped product Einstein metrics, Comm. Anal. Geom.20(2012), 271–311

    work page 2012

  26. [26]

    Math.18(2014), 159–190

    He, C., Petersen, P., and Wylie, W.,Warped product Einstein metrics over spaces with constant scalar curvature, Asian J. Math.18(2014), 159–190

    work page 2014

  27. [27]

    Hitchin, N.,Compact Four-Dimensional Einstein Manifolds, J. Differ. Geom.9(1974), 435–441

    work page 1974

  28. [28]

    and Kim, Y

    Kim, D.-S. and Kim, Y. H.,Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc.131(2003), 2573–2576

    work page 2003

  29. [29]

    Sym- pos

    Koiso, N.,On Rotationally Symmetric Hamilton’s Equation for K¨ ahler–Einstein Metrics, Proc. Sym- pos. Pure Math.54(1993), 327–337

    work page 1993

  30. [30]

    Math.95(2010), 191–199

    Limoncu, M.,Modifications of the Ricci tensor and applications, Arch. Math.95(2010), 191–199

    work page 2010

  31. [31]

    N., and Pope, C

    L¨ u, H., Page, D. N., and Pope, C. N.,New inhomogeneous Einstein metrics on sphere bundles over Einstein-K¨ ahler manifolds, Phys. Lett. B593(2004), 218–226

    work page 2004

  32. [32]

    Ma, L.,Remarks on compact shrinking Ricci solitons of dimension four, C. R. Math. Acad. Sci. Paris 351(2013), 817–823

    work page 2013

  33. [33]

    and Rimoldi, M.,Some triviality results for quasi-Einstein manifolds and Einstein warped products, Geom

    Mastrolia, P. and Rimoldi, M.,Some triviality results for quasi-Einstein manifolds and Einstein warped products, Geom. Dedic.169(2014), 225–237

    work page 2014

  34. [34]

    Qian, Z.,Estimates for weighted volumes and applications, Quart. J. Math.48(1997), 235–242

    work page 1997

  35. [35]

    and Tenenblat, K.,Noncompact quasi-Einstein manifolds conformal to a Euclidean space, Math

    Ribeiro Jr., E. and Tenenblat, K.,Noncompact quasi-Einstein manifolds conformal to a Euclidean space, Math. Nachr.294(2021), no. 1, 132–144

    work page 2021

  36. [36]

    Math.252(2011), 207–218

    Rimoldi, M.,A remark on Einstein warped products, Pacific J. Math.252(2011), 207–218

    work page 2011

  37. [37]

    Seshadri, H.,Weyl curvature and the Euler characteristic in dimension four, Differ. Geom. Appl.24 (2006), 172–177

    work page 2006

  38. [38]

    A., Mandal, P., and Mondal, C

    Shaikh, A. A., Mandal, P., and Mondal, C. K.,Diameter estimation of(m, p)-quasi Einstein manifolds, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci.94(2024), 513–518

    work page 2024

  39. [39]

    Tadano, H.,m-Bakry– ´Emery Ricci curvatures, Riccati inequalities, and bounded diameters, Differ. Geom. Appl.80(2022), 101832

    work page 2022

  40. [40]

    Tadano, H.,An upper diameter bound for compact Ricci solitons with application to the Hitchin–Thorpe inequality. II, J. Math. Phys.59(2018), no. 4, 043507, 3

    work page 2018

  41. [41]

    Tadano, H.,An improvement of the Myers theorem viam-Bakry– ´Emery Ricci curvature withε-range, Ann. Mat. Pura Appl. (4)204(2025), no. 6, 2757–2769

    work page 2025

  42. [42]

    Tadano, H.,Myers-type theorems viam-Bakry– ´Emery Ricci curvature of quadratic decays, Internat. J. Math.36(2025), no. 14, 2450059

    work page 2025

  43. [43]

    Tadano, H.,New Compactness Criteria via Integral Radialm-Bakry– ´Emery Ricci Curvatures, Bull. Sci. Math. (2026), 103837

    work page 2026

  44. [44]

    Tadano, H.,Some Cheeger–Gromov–Taylor type compactness theorems viam-Bakry– ´Emery andm- modified Ricci curvatures, Nonlinear Anal.199(2020), 112045

    work page 2020

  45. [45]

    A.,Some remarks on the Gauss-Bonnet integral, J

    Thorpe, J. A.,Some remarks on the Gauss-Bonnet integral, J. Math. Mech.18(1969), 779–786

    work page 1969

  46. [46]

    Math.254(2011), 449–464

    Wang, L.,On noncompactm-quasi-Einstein metrics, Pacific J. Math.254(2011), 449–464

    work page 2011

  47. [47]

    F.,Diameter estimate for compact quasi-Einstein metrics, Math

    Wang, L. F.,Diameter estimate for compact quasi-Einstein metrics, Math. Z.273(2013), 801–809

    work page 2013

  48. [48]

    F.,Potential function estimates for quasi-Einstein metrics, J

    Wang, L. F.,Potential function estimates for quasi-Einstein metrics, J. Funct. Anal.267(2014), 1986–2004

    work page 2014

  49. [49]

    and Zhu, X.,K¨ ahler–Ricci solitons on toric manifolds with positive first Chern class, Adv

    Wang, X.-J. and Zhu, X.,K¨ ahler–Ricci solitons on toric manifolds with positive first Chern class, Adv. Math.188(2004), 87–103

    work page 2004

  50. [50]

    and Wylie, W.,Comparison Geometry for the Smooth Metric Measure Spaces, Proc

    Wei, G. and Wylie, W.,Comparison Geometry for the Smooth Metric Measure Spaces, Proc. 4th Int. Congr. Chin. Math. (ICCM)II(2007), 1–4

    work page 2007

  51. [51]

    Wu, J.-Y.,Myers’ type theorem with the Bakry- ´Emery Ricci tensor, Ann. Glob. Anal. Geom.54(2018), 541–549

    work page 2018

  52. [52]

    Zhang, Y.,A note on the Hitchin–Thorpe inequality and Ricci flow on 4-manifolds, Proc. Amer. Math. Soc.140(2012), no. 5, 1777–1783. Departamento de Matem´atica, Universidade Federal do Cear ´a - UFC, Campus do Pici, Av. Humberto Monte, Bloco 914, 60455-760, Fortaleza - CE, Brazil Email address:sanetto@alu.ufc.br

    work page 2012