L^2-instability of the Taub-Bolt metric under the Ricci flow

TL;DR

This paper demonstrates that small L^2 perturbations of the Ricci-flat Taub-Bolt metric can evolve under Ricci flow into finite-time singularities modeled on a known shrinking soliton, revealing instability properties.

Contribution

It establishes the L^2-instability of the Taub-Bolt metric under Ricci flow and constructs explicit perturbations leading to singularity formation.

Findings

Existence of L^2-small perturbations leading to singularities

Construction of Ricci flows with prescribed singularity models

Extension of previous methods to the Taub-Bolt setting

Abstract

In this paper we prove that there exists a compact perturbation of the Ricci flat Taub-Bolt metric that evolves under the Ricci flow into a finite time singularity modelled on the shrinking solition FIK [5]. Moreover, this perturbation can be made arbitrarily L^2-small with respect to the Taub-Bolt metric. The method of proof closely follows the strategy adopted in [14], where the author constructs, via a Wa\.zewski box argument, Ricci flows on compact manifolds which encounter finite singularities modelled on a given asymptotically conical shrinking soliton.