Paper · Aug 26, 2026

Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

JeongHyeong Park, Wooseok Shin

math.DGarXiv:2608.25704
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation for the soliton potential . Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.

View PDF View HTML arXiv page
Submitted
Aug 26, 2026
Subjects
math.DG

Comments: 6 pages

Ricci solitons4-manifoldsLimit spaces & RCDHomogeneous spacesStability & uniquenessBoundaries & non-compact

Its neighbourhood

Connected papers

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 SU(2)-invariant steady gradient Ricci solitons on four-manifolds (Timothy Buttsworth, 2021)1Buttsworth 2021 Rigidity of four-dimensional Kähler-Ricci solitons (Xiaodong Cao, Ernani Ribeiro and Hung Tran, 2022)7Cao 2022 On curvature estimates for four-dimensional gradient Ricci solitons (Huai-Dong Cao, 2025)8Cao 2025 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)3Ayad 2026 Four-dimensional Lorentzian algebraic Ricci solitons (Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo, 2026)6Garcia-Rio 2026 A note on the classification of four-dimensional gradient steady and expanding Ricci solitons (Huai-Dong Cao and Junming Xie, 2026)2Cao 2026 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)5Wang 2026 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)4Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. -invariant steady gradient Ricci solitons on four-manifolds

    Timothy Buttsworth

    Topics 4-manifolds, Homogeneous spaces, Boundaries & non-compact · Shared terms “four manifolds”, “four dimensional”, “gradient Ricci”

  2. A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

    Huai-Dong Cao and Junming Xie

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “four dimensional”

  3. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”

  4. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “four dimensional”, “gradient Ricci”, “gradient shrinking”

  5. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Boundaries & non-compact, Stability & uniqueness · Shared terms “four dimensional”, “gradient Ricci”, “gradient shrinking”

  6. Four-dimensional Lorentzian algebraic Ricci solitons

    Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey and Ramon Vazquez-Lorenzo

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “four dimensional”

  7. Rigidity of four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran

    Topics 4-manifolds, Stability & uniqueness, Ricci solitons · Shared terms “four dimensional”, “soliton structure”, “gradient shrinking”

  8. On curvature estimates for four-dimensional gradient Ricci solitons

    Huai-Dong Cao

    Topics 4-manifolds, Ricci solitons · Shared terms “four dimensional”, “gradient Ricci”

Read around it

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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.