James Waldron

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.

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.