Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

Abstract

Using a stability criterion due to Kröncke, we show, providing $${n\ne 2k}$$ , the Kähler–Einstein metric on the Grassmannian $$Gr_{k}(\mathbb {C}^{n})$$ 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.