Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

TL;DR

This paper demonstrates the dynamical instability of the K"ahler--Einstein metric on Grassmannians under Ricci flow, extending previous results and highlighting limitations of Kr"oncke's stability criterion on other symmetric spaces.

Contribution

It shows the instability of Grassmannian metrics via coadjoint orbit techniques and reveals the method's limitations for other Hermitian symmetric spaces.

Findings

Grassmannian K"ahler--Einstein metrics are dynamically unstable under Ricci flow.

Kr"oncke's stability criterion does not apply to other compact Hermitian symmetric spaces.

The Grassmannian can be described as a coadjoint orbit of SU(n).

Abstract

Using a stability criterion due to Kr\"oncke, we show, providing ${n\neq 2k}$, the K\"ahler--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 Kr\"oncke 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 Kr\"oncke's method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.