Paper · May 29, 2025

Rotationally symmetric Ricci Flow on

Ming Hsiao

math.DGarXiv:2505.23157
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Abstract

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.

Comments: 20 pages. All comments are welcome

Ricci flowSingularities(Non)collapsingWeak & singular flowsHarmonic maps & DeTurck trickRotational symmetry & warped products

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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 Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions (Kin Ming Hui, 2021)4Hui 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)2Chen 2022 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)5Bahuaud 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)7Lee 2024 Uniqueness of Ricci flow with scaling invariant estimates (Man-Chun Lee, 2025)3Lee 2025 Steady soliton with 𝓛¹ decay curvature (Ming Hsiao, 2026)8Hsiao 2026 On the uniqueness of Ricci flows from certain Alexandrov spaces (Laura Flower, 2026)6Flower 2026 An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow (Hongxin Guo and Guozhi Lin, 2026)1Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

    Hongxin Guo and Guozhi Lin PDF

    Topics Rotational symmetry & warped products, Singularities · Shared terms “symmetric Ricci”, “rotationally symmetric”, “invariant curvature”

  2. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows PDF

    Topics Rotational symmetry & warped products, Weak & singular flows, Singularities · Shared terms “rotationally symmetric”, “short time”

  3. Uniqueness of Ricci flow with scaling invariant estimates

    Man-Chun Lee PDF

    Topics Harmonic maps & DeTurck trick, (Non)collapsing · Shared terms “scaling invariant”, “complete Ricci”, “invariant curvature”

  4. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui PDF

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  5. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Harmonic maps & DeTurck trick, Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”

  6. On the uniqueness of Ricci flows from certain Alexandrov spaces

    Laura Flower PDF

    Topics (Non)collapsing, Weak & singular flows · Shared terms “scaling invariant”, “invariant curvature”, “curvature bounds”

  7. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu PDF

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Singularities · Shared terms “existence theory”, “short time”, “time existence”

  8. Steady soliton with decay curvature

    Ming Hsiao PDF

    Topics (Non)collapsing · Same author Ming Hsiao · Shared terms “curvature bounds”, “noncollapsed”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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
    κ-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.