Paper · Jul 16, 2025

Hamilton's identity and rigidity of complete gradient solitons

Antonio W. Cunha, Antonio N. Silva, William Wylie

math.DGarXiv:2507.12381
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Abstract

In this work, we study gradient solitons to general geometric flows. Our approach is to understand what assumptions need to be made about a flow in order to extend results about Ricci solitons. In this direction, we identify an identity, first exploited in the pioneering work of Richard Hamilton in the case of Ricci solitons, which we call Hamilton's identity. We show that a version of this identity for an arbitrary geometric flow allows one to recover results about rigidity, the growth of the potential function, volume growth and the Omori-Yau maximum principle that have been proven for gradient Ricci solitons.

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Submitted
Jul 16, 2025
Subjects
math.DG

Comments: 24 pages

Ricci solitonsStability & uniqueness

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 Volume growth estimates of gradient Ricci solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2022)4Chan 2022 Rigidity results for shrinking and expanding Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)2Leandro 2022 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)3Cao 2024 Some rigidity results on shrinking gradient Ricci soliton (Jianyu Ou, Yuanyuan Qu and Guoqiang Wu, 2024)1Ou 2024 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)5Li 2024 Rigidity of Five-Dimensional shrinking gradient Ricci solitons (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2025)7Li 2025 Ricci-harmonic flow of G₂ and Spin(7)-structures (Shubham Dwivedi, 2026)6Dwivedi 2026 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu

    Topics Stability & uniqueness, Ricci solitons · Shared terms “gradient Ricci”, “maximum principle”

  2. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Stability & uniqueness, Ricci solitons · Shared terms “potential function”

  3. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran

    Topics Stability & uniqueness, Ricci solitons · Shared terms “gradient Ricci”, “Richard”, “Hamilton”

  4. Volume growth estimates of gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Ricci solitons · Shared terms “volume growth”, “gradient Ricci”

  5. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Stability & uniqueness, Ricci solitons · Shared terms “gradient Ricci”, “potential function”

  6. Ricci-harmonic flow of and Spin(7)-structures

    Shubham Dwivedi

    Topics Stability & uniqueness, Ricci solitons · Shared terms “Hamilton identity”, “gradient Ricci”

  7. Rigidity of Five-Dimensional shrinking gradient Ricci solitons

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Stability & uniqueness, Ricci solitons · Shared terms “gradient Ricci”

  8. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu

    Topics Stability & uniqueness, Ricci solitons · Shared terms “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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.