Rigidity of SUn-Type Symmetric Spaces

Abstract

Abstract We prove that the bi-invariant Einstein metric on $SU_{2n+1}$ 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 $\mathbb{C}\mathbb{P}^{2n}\times \mathbb{C}\mathbb{P}^{1}$ 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.