Ancient and expanding spin ALE Ricci flows

TL;DR

This paper classifies certain ancient and expanding Ricci flows on spin ALE manifolds, revealing that under specific conditions, they are hyperkähler ALE metrics, and introduces a new functional controlling their large-scale behavior.

Contribution

It provides a classification of spin ALE ancient Ricci flows and expanding solitons, and introduces a renormalized functional related to weighted mass for ALE orbifolds.

Findings

Only hyperkähler ALE metrics arise as spin ancient Ricci flows with $SU(2)$ groups at infinity.

The large-scale behavior of Perelman's $d$-functional is controlled by a new $d$-functional related to weighted mass.

The classification extends understanding of Ricci flows on non-compact manifolds with specific symmetry and decay conditions.

Abstract

We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in $SU(2)$ and mild decay at infinity are hyperk\"ahler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's $\mu$-functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized $\lambda_{\mathrm{ALE}}$-functional related to a notion of weighted mass.