Destabilising compact warped product Einstein manifolds

TL;DR

This paper investigates the stability of warped product Einstein metrics under Ricci flow, revealing instability in low dimensions and introducing a new destabilizing perturbation to analyze high-dimensional cases.

Contribution

It generalizes previous stability results and introduces the Ricci variation as a new tool to study the instability of warped product Einstein metrics.

Findings

All warped product Einstein metrics are unstable in low dimensions.

Certain infinite families are unstable in high dimensions.

The Ricci variation is a new destabilizing perturbation.

Abstract

The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics and Ricci solitons, we introduce a new destabilising perturbation (the Ricci variation) and show that certain infinite families of warped product Einstein metrics will be unstable in high dimensions.