Stability of compact symmetric spaces (Q2117484)

The article is a study on the stability problem (in the sense of Koiso) for the Einstein-Hilbert functional on compact symmetric spaces. The main result that clarifies the stability status of quaternionic and Cayley projective planes is the following: Theorem. The Cayley projective plane $\mathbf{O}P^2=F_4/\mathrm{Spin}(9)$ is stable in the sense of Koiso. The quaternionic Grassmannians $\mathbf{Gr}_r H^{r+s}=\mathrm{Sp}(r+s)/\mathrm{Sp}(r)\times \mathrm{Sp}(s)$ of quaternionic subspaces of dimension $r$ in $H^{r+s}$ are unstable in the sense of Koiso for all parameters $r,s\geq2$, but they are stable for $r=1$ or $s=1$. In particular, $HP^2=\mathbf{Gr}_1H^3$ is stable.