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.

中文翻译：

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破坏紧凑翘曲积爱因斯坦流形

研究了作为 Ricci 流不动点的翘曲积爱因斯坦度量的线性稳定性。我们概括了 Gibbons、Hartnoll 和 Pope 的结果，并表明在足够低的维度下，所有扭曲的产品爱因斯坦指标都是不稳定的。通过利用扭曲积爱因斯坦度量、准爱因斯坦度量和 Ricci 孤子之间的关系，我们引入了一种新的不稳定扰动（Ricci 变体），并表明某些无限的扭曲积爱因斯坦度量在高维上是不稳定的。