Communications in Analysis and Geometry

Volume 29 (2021)

Number 5

Destabilising compact warped product Einstein manifolds

Pages: 1061 – 1094



Wafaa Batat (École Nationale Polytechnique d’Oran, Algeria)

Stuart Hall (School of Mathematics and Statistics, Newcastle University, Newcastle-upon-Tyne, United Kingdom)

Thomas Murphy (Department of Mathematics, California State University, Fullerton, Calif., U.S.A.)


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.

Received 27 September 2017

Accepted 19 December 2018

Published 1 December 2021