Paper · Oct 21, 2024

Orbifold singularity formation along ancient and immortal Ricci flows

Alix Deruelle, Tristan Ozuch

math.DGarXiv:2410.16075v2
PDF Abstract page

As on arXiv

Abstract

In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension with Einstein orbifolds as tangent flows at infinity. For instance, for any , we obtain continuous families of non-isometric ancient Ricci flows on depending on a number of parameters growing linearly in , and a family of half-PIC ancient Ricci flows on . The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics.

Comments: 160 pages, no figure, v2: added initial condition in some statements in the immortal case, proofs are unchanged

Ricci flowRicci solitonsAncient solutionsRicci-flat & Calabi–YauSingularitiesSphere theoremsStability & uniquenessLong-time existence & convergence

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 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)7Lynch 2022 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)8Deruelle 2024 Ancient and expanding spin ALE Ricci flows (Isaac M. Lopez and Tristan Ozuch, 2024)1Lopez 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)6Deng 2025 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)2Kim 2025 Linear stability and instability of Kähler Ricci solitons (Keaton Naff and Tristan Ozuch, 2025)3Naff 2025 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)4Deng 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)5Daskalopoulo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient and expanding spin ALE Ricci flows

    Isaac M. Lopez and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions, Ricci solitons · Same author Tristan Ozuch · Shared terms “ALE”, “orbifold”

  2. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Same author Tristan Ozuch · Shared terms “along Ricci”, “linear stability”, “unstable”

  3. Linear stability and instability of Kähler Ricci solitons

    Keaton Naff and Tristan Ozuch PDF

    Topics Singularities, Stability & uniqueness, Ricci solitons · Same author Tristan Ozuch · Shared terms “orbifold singularities”, “linear stability”, “real dimension”

  4. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics Ricci-flat & Calabi–Yau, Ancient solutions, Singularities · Shared terms “tangent flow”, “dichotomy”

  5. Unique asymptotics of symmetric ancient ovals of Ricci flow

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

    Topics Sphere theorems, Ancient solutions, Singularities · Shared terms “instance”, “tangent flow”

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

    Yuxing Deng PDF

    Topics Sphere theorems, Ancient solutions, Singularities · Shared terms “orbifold”

  7. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego PDF

    Topics Sphere theorems, Ancient solutions, Singularities · Shared terms “singularity formation”, “PIC”

  8. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics Sphere theorems · Same author Alix Deruelle · 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 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.