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Orbifold singularity formation along ancient and immortal Ricci flows

Alix Deruelle, Tristan Ozuch

TL;DR

This work delivers a parabolic gluing framework to produce large families of ancient and immortal Ricci flows in dimension four that develop orbifold singularities whose tangent models are Einstein orbifolds. Central to the construction are Kronheimer’s hyperkähler ALE spaces glued to spherical orbifolds, with a novel linear stability notion based on the selfdual curvature $\mathbf{R}^+$ guiding which singularities persist or desingularize via bubbling of Ricci-flat ALE metrics. The authors develop a robust analytic apparatus—first approximate solutions, an approximate kernel capturing gluing directions, Liouville-type rigidity theorems, and a linear-constrained fixed-point scheme—together with intricate projections and Schauder estimates in weighted Hölder spaces. The resulting ancient and immortal flows exhibit controlled curvature decay, local smooth convergence away from orbifold points, and Gromov-Hausdorff convergence to the underlying Einstein orbifold, confirming that Ricci flow can spontaneously desingularize certain orbifold singularities and, in the immortal case, realize a broad class of orbifold limits as long-time attractors.

Abstract

In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in\mathbb{N}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k(\mathbb{S}^2\times \mathbb{S}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on $\mathbb{CP}^2\#\mathbb{CP}^2$. 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.

Orbifold singularity formation along ancient and immortal Ricci flows

TL;DR

This work delivers a parabolic gluing framework to produce large families of ancient and immortal Ricci flows in dimension four that develop orbifold singularities whose tangent models are Einstein orbifolds. Central to the construction are Kronheimer’s hyperkähler ALE spaces glued to spherical orbifolds, with a novel linear stability notion based on the selfdual curvature guiding which singularities persist or desingularize via bubbling of Ricci-flat ALE metrics. The authors develop a robust analytic apparatus—first approximate solutions, an approximate kernel capturing gluing directions, Liouville-type rigidity theorems, and a linear-constrained fixed-point scheme—together with intricate projections and Schauder estimates in weighted Hölder spaces. The resulting ancient and immortal flows exhibit controlled curvature decay, local smooth convergence away from orbifold points, and Gromov-Hausdorff convergence to the underlying Einstein orbifold, confirming that Ricci flow can spontaneously desingularize certain orbifold singularities and, in the immortal case, realize a broad class of orbifold limits as long-time attractors.

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.

Paper Structure

This paper contains 95 sections, 106 theorems, 607 equations.

Table of Contents

  1. Introduction
  2. Main motivations
  3. Noncollapsed Ancient Ricci flows
  4. Degeneration of Einstein $4$-manifolds and obstructions
  5. Limits of thick immortal Ricci flows
  6. Main results: Dynamical (in)stability of Einstein $4$-orbifolds
  7. Related results
  8. Brendle-Kapouleas construction: links and differences with our work
  9. Other orbifold Ricci flows
  10. Further directions
  11. Discussion of the proofs
  12. Organization of the article
  13. From obstructions to Ricci flows
  14. Basics of $4$-dimensional geometry and notations
  15. Notations
  16. ...and 80 more sections

Key Result

Theorem 1.3

Let $\Gamma\subset \operatorname{SU}(2)$ be a finite group acting freely on $\mathbb{S}^3$ and let $(M_o^4,g_o)$ be a metric suspension over $\mathbb{S}^3/\Gamma$ endowed with a metric of constant curvature with Einstein constant $\Lambda$. Let $(N^4,g_b)$ be a hyperkähler ALE metric asymptotic to $ such that: In particular, any hyperkähler ALE metric can be bubbled-off that way in the past of an

Theorems & Definitions (277)

  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Conjecture 1.12
  • Conjecture 1.13
  • Conjecture 1.14
  • Conjecture 1.15
  • ...and 267 more