Paper · Nov 15, 2023

Unique Asymptotics of Steady Ricci Solitons with Symmetry

Zilu Ma, Hamidreza Mahmoudian, Natasa Sesum

math.DGarXiv:2311.09405
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Abstract

In this paper we study 4d gradient steady Ricci solitons, which are weak -solutions, and admit O(3)-symmetry. Under a weak curvature decay condition, we find precise geometric asymptotics of such solitons, which are similar to those for 3d compact -solutions found in [ABDS22]. This is the first step towards the classification of 4d gradient steady Ricci solitons and more general ancient Ricci flows.

Ricci flowRicci solitonsAncient solutions3-manifolds4-manifolds

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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 On a dichotomy of the curvature decay of steady Ricci soliton (Pak-Yeung Chan and Bo Zhu, 2021)7Chan 2021 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)5Lai 2022 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)8Chan 2023 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)3Isenberg 2025 On curvature estimates for four-dimensional gradient Ricci solitons (Huai-Dong Cao, 2025)4Cao 2025 Asymptotic Geometry of Four-Dimensional Steady Solitons (Aprameya Girish Hebbar and Natasa Sesum, 2026)2Hebbar 2026 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)1Daskalopoulo 2026 O(3)-symmetry of four-dimensional steady Ricci solitons (Hanbing Fang, 2026)6Fang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al. PDF

    Topics Ancient solutions · Same author Natasa Sesum · Shared terms “unique asymptotics”

  2. Asymptotic Geometry of Four-Dimensional Steady Solitons

    Aprameya Girish Hebbar and Natasa Sesum PDF

    Topics 4-manifolds, Ancient solutions, Ricci solitons · Same author Natasa Sesum · Shared terms “curvature decay”

  3. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al. PDF

    Topics 4-manifolds, Ricci solitons · Same authors Zilu Ma, Natasa Sesum

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

    Huai-Dong Cao PDF

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

  5. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai PDF

    Topics Ancient solutions, 3-manifolds, Ricci solitons

  6. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang PDF

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

  7. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “steady Ricci”, “curvature decay”

  8. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics 4-manifolds, Ricci solitons · Same author Zilu Ma

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 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.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.