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On $κ$-solutions and canonical neighborhoods in 4d Ricci flow

Robert Haslhofer

Abstract

We introduce a classification conjecture for $κ$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of $\mathbb{Z}_2^2\times \mathrm{O}_3$-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

On $κ$-solutions and canonical neighborhoods in 4d Ricci flow

Abstract

We introduce a classification conjecture for -solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of -symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.
Paper Structure (3 sections, 2 theorems, 15 equations)

This paper contains 3 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Construction of new $\kappa$-solutions
  3. Canonical neighborhoods

Key Result

Theorem 1.2

On $S^4$, there exists a 1-parameter family of $\mathbb{Z}_2^2\times\mathrm{O}_3$-symmetric $\kappa$-solutions, whose tangent flow at $-\infty$ is a round shrinking $\mathbb{R}^2\times S^2$.

Theorems & Definitions (13)

  • Definition 1.1: $\kappa$-solution
  • Theorem 1.2: bubble-sheet ovals
  • Conjecture 1.3: $\kappa$-solutions in 4d Ricci flow
  • Definition 1.4: metric Ricci flow
  • Definition 1.5: cylindrical singularity
  • Definition 1.6: ancient asymptotically cylindrical 4d Ricci flow
  • Conjecture 1.7: ancient 4d Ricci flows
  • Theorem 1.8: canonical neighborhoods
  • Conjecture 1.9: non-uniqueness
  • proof : Proof of Theorem \ref{['thm_ovals']}
  • ...and 3 more