# Rigidity and instability of SU(n) symmetric spaces

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Focusing on the Lie group SU(n) and associated symmetric spaces,

I investigate two related topics. The main result is 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-Kaehler, non-product example of this phenomenon adding to the famous

example of Koiso from the eighties. I also explore the relationship between

the question of rigidity and instability (under the Ricci flow) of Einstein

metrics, and present results in this direction for complex Grassmannians.