Paper · Apr 22, 2026

Steady soliton with decay curvature

Ming Hsiao

math.DGarXiv:2604.20224
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Abstract

In this paper, we establish a compactness theorem for gradient Ricci solitons with scalar curvature bounds and uniform lower bounds of harmonic coordinates. Our approach is to bootstrap regularity in harmonic coordinates by exploiting the soliton equation. As an application, we show that the regular part of any noncollapsed limit of gradient Ricci solitons with bounded Ricci curvature is smooth. Further, we show that a steady gradient Ricci soliton is asymptotically cylindrical under an -decay assumption on its Ricci curvature.

Comments: 14 pages

Ricci solitons(Non)collapsingScalar curvatureLimit spaces & RCD

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 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)4Chan 2022 Notes on scalar curvature lower bounds of steady gradient Ricci solitons (Shota Hamanaka, 2024)1Hamanaka 2024 On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature (Yuxing Deng and Yuehan Hao, 2024)6Deng 2024 Non-collapsing of Ricci shrinkers with bounded curvature (Conghan Dong and Yu Li, 2024)8Dong 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)7Deng 2025 On the structure of noncollapsed Ricci flow limit spaces (Hanbing Fang and Yu Li, 2025)2Fang 2025 Curvature estimates for steady and expanding solitons in higher dimensions (Pak-Yeung Chan and Ming Hsiao, 2025)5Chan 2025 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka PDF

    Topics Limit spaces & RCD, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “lower bound”

  2. On the structure of noncollapsed Ricci flow limit spaces

    Hanbing Fang and Yu Li PDF

    Topics (Non)collapsing, Limit spaces & RCD · Shared terms “regular part”, “compactness theorem”, “noncollapsed”

  3. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics (Non)collapsing, Limit spaces & RCD, Scalar curvature · Shared terms “lower bound”, “uniform lower”

  4. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al. PDF

    Topics Limit spaces & RCD, Scalar curvature, Ricci solitons · Shared terms “steady solitons”, “compactness theorem”, “lower bound”

  5. Curvature estimates for steady and expanding solitons in higher dimensions

    Pak-Yeung Chan and Ming Hsiao PDF

    Topics Ricci solitons · Same author Ming Hsiao · Shared terms “steady solitons”, “gradient Ricci”, “curvature bounds”

  6. On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

    Yuxing Deng and Yuehan Hao PDF

    Topics (Non)collapsing, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “noncollapsed”

  7. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng PDF

    Topics (Non)collapsing, Scalar curvature, Ricci solitons · Shared terms “gradient Ricci”, “steady gradient”, “noncollapsed”

  8. Non-collapsing of Ricci shrinkers with bounded curvature

    Conghan Dong and Yu Li PDF

    Topics (Non)collapsing, Limit spaces & RCD, Ricci solitons · Shared terms “curvature bounds”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's Îș-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    Îș-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.