Paper · Apr 22, 2026

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

Samuel Belo

math.DGarXiv:2604.21002v2
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Abstract

We study compact -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 -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.

Comments: 16 pages. Comments are welcome

Ricci solitonsEinstein metricsAlmost & η-Ricci solitons4-manifoldsLimit spaces & RCD

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2022)1Cheng 2022 The extended quasi-Einstein manifolds with generalised Ricci solitons (Zhiming Huang, Weijun Lu and Fuhong Su, 2022)2Huang 2022 Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four (Cameron MacMahon, 2024)5MacMahon 2024 A Characterization of Quasi-Einstein Metrics (Antonio Airton Freitas Filho, 2025)8Filho 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)3Bhattacharya 2025 On a compact Einstein-type manifolds with Riemannian foliations (Jungwoo Moon, 2026)7Moon 2026 The rigity of spaces with cyclic parallel Ricci tensor (Fengyuan Zhang, 2026)6Zhang 2026 Direct proofs for spaces with cyclic parallel Ricci tensor (Fengyuan Zhang, 2026)4Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

    Xu Cheng, Ernani Ribeiro and Detang Zhou PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “Thorpe inequality”, “Hitchin Thorpe”, “dimensional compact”

  2. The extended quasi-Einstein manifolds with generalised Ricci solitons

    Zhiming Huang, Weijun Lu and Fuhong Su PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  3. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  4. Direct proofs for spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “compact quasi”

  5. Topological Obstructions to the Existence of Compact Shrinking Ricci Solitons in Dimension Four

    Cameron MacMahon PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “Thorpe inequality”, “Hitchin Thorpe”, “dimension four”

  6. The rigity of spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  7. On a compact Einstein-type manifolds with Riemannian foliations

    Jungwoo Moon PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons

  8. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.