Stuart James Hall

We prove that the bi-invariant Einstein metric on $S{U}_{2n+1}$ is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of ${\mathbb{CP}}^{2n}\times {\mathbb{CP}}^1$ found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler–Einstein metrics on `odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.

In this note we show that the bi-invariant Einstein metric on the compact Lie group $G_2$ is dynamically unstable as a fixed point of the Ricci flow. This completes the stability analysis for the bi-invariant metrics on the compact, connected, simple Lie groups. Interestingly, $G_2$ is the only unstable exceptional group.

Using a stability criterion due to Kröncke, we show, providing $n\ne 2k$, the Kähler–Einstein metric on the Grassmannian $G{r}_k({\mathbb{C}}^n)$ of complex $k$-planes in an $n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on ${\mathbb{CP}}^n$ for $n>1$. The key to the proof is using the description of Grassmannians as certain coadjoint orbits of $SU(n)$. We are also able to prove that Kröncke's method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

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.