Orbifold singularity formation along ancient and immortal Ricci flows

TL;DR

This paper constructs numerous ancient and immortal Ricci flows in four dimensions with Einstein orbifolds as tangent flows at infinity, revealing new phenomena in higher-dimensional Ricci flow singularity formation.

Contribution

It introduces new examples of Ricci flows with orbifold singularities in four dimensions and explores their stability and formation mechanisms.

Findings

Constructed families of ancient Ricci flows on connected sums of spheres.

Identified stability conditions for orbifold singularities in Ricci flows.

Proposed conjecture on the non-occurrence of certain unstable orbifold solitons.

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.