Paper · Dec 6, 2022

Rigidity of the Bryant Ricci soliton

Ziyi Zhao, Xiaohua Zhu

math.DGarXiv:2212.02889v2
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Abstract

We introduce a new curvature-pinching condition, which is weaker than the positive sectional curvature or PIC1, and then we prove several rigidity results for the rotationally symmetric solutions of steady Ricci solitons, i.e., the Bryant Ricci solitons.

Comments: Some parts in Section 1 are revised

Ricci solitonsCurvature pinchingSphere theoremsStability & uniquenessRotational symmetry & warped products

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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 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2023)1Zhao 2023 Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set (Ziyi Zhao and Xiaohua Zhu, 2024)2Zhao 2024 No compact split limit Ricci flow of type II from the blow-down (Ziyi Zhao and Xiaohua Zhu, 2024)3Zhao 2024 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)8Chen 2025 Curvature pinching of asymptotically conical gradient expanding Ricci solitons (Huai-Dong Cao and Junming Xie, 2025)4Cao 2025 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)6Daskalopoulo 2026 Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching (Jian Ge, 2026)7Ge 2026 Sectional Curvature Pinching of Two-Step Nilmanifolds (Tomoya Tatsuno, 2026)5Tatsuno 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. steady gradient Ricci solitons with nonnegative curvature away from a compact set

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “Bryant Ricci”, “sectional curvature”

  2. Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu · Shared terms “Bryant Ricci”, “steady Ricci”

  3. No compact split limit Ricci flow of type II from the blow-down

    Ziyi Zhao and Xiaohua Zhu PDF

    Topics Ricci solitons · Same authors Ziyi Zhao, Xiaohua Zhu

  4. Curvature pinching of asymptotically conical gradient expanding Ricci solitons

    Huai-Dong Cao and Junming Xie PDF

    Topics Curvature pinching, Sphere theorems, Ricci solitons · Shared terms “positive sectional”, “steady Ricci”, “sectional curvature”

  5. Sectional Curvature Pinching of Two-Step Nilmanifolds

    Tomoya Tatsuno PDF

    Topics Curvature pinching, Stability & uniqueness, Ricci solitons · Shared terms “sectional curvature”

  6. Unique asymptotics of symmetric ancient ovals of Ricci flow

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

    Topics Sphere theorems, Stability & uniqueness · Same author Ziyi Zhao

  7. Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching

    Jian Ge PDF

    Topics Curvature pinching · Shared terms “pinching condition”, “PIC”, “sectional curvature”

  8. On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

    Zhengnan Chen PDF

    Topics Curvature pinching, Sphere theorems, Ricci solitons · Shared terms “PIC”

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 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. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.