Paper · Feb 9, 2023

Ancient Ricci flows of bounded girth

Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente, Mat Langford

math.DGarXiv:2302.04964v2Orbita Math. 3 (2026) 185-213
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Abstract

For each , we construct a 'pancake-like', -invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in dimension three. The construction hinges on the Ricci flow invariance of certain conditions on the curvature and its spatial derivatives under this symmetry regime, whose proof does not follow from Hamilton's tensor maximum principle.

Comments: 31 pages, no figures. In this second version, we corrected the maximum principle proofs of Section 3.2

Ricci flowAncient solutions3-manifoldsSphere theorems

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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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)2Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)1Bourni 2022 Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2022)6Bourni 2022 Ancient pancake solutions to fully nonlinear curvature flows (Sathya Rengaswami and Mat Langford, 2023)5Rengaswami 2023 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)3Zhang 2024 Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry (Lucas Lavoyer and Luke T. Peachey, 2025)8Lavoyer 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)4Daskalopoulo 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)7Bang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford

  2. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford

  3. A rigidity result for ancient Ricci flows

    Qi S. Zhang PDF

    Topics Ancient solutions · Shared terms “backward”

  4. Unique asymptotics of symmetric ancient ovals of Ricci flow

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

    Topics Sphere theorems, Ancient solutions · Shared terms “invariant ancient”, “positive curvature”, “curvature operator”

  5. Ancient pancake solutions to fully nonlinear curvature flows

    Sathya Rengaswami and Mat LangfordWider flows PDF

    Topics Ancient solutions · Same author Mat Langford · Shared terms “invariant ancient”

  6. Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows PDF

    Same authors Theodora Bourni, Mat Langford · Shared terms “Hamilton”

  7. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du PDF

    Topics Ancient solutions · Shared terms “backward”

  8. Steady Gradient Ricci Solitons with Symmetry

    Lucas Lavoyer and Luke T. Peachey PDF

    Topics Sphere theorems · Shared terms “positive curvature”, “curvature operator”

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) 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) 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)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  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.