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

2018/06/05 by Stuart James Hall, Thomas Murphy, Hall, Stuart James +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1806.01722

21 pages, v3 minor rewrites to clarify arguments, to appear in J. Geom. Anal

arxiv created 2020/09/20 · arxiv updated 2020/09/22

Abstract

Using a stability criterion due to Kröncke, we show, providing n≠ 2k, the Kähler--Einstein metric on the Grassmannian Gr_k(ℂⁿ) 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 \mathbbCPⁿ 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.

Related

• Dynamic instability of \mathbbCP^N under Ricci flow
• Kähler Ricci flow with vanished Futaki invariant
• Stability on Kähler-Ricci flow, I
• Linear and dynamical stability of Ricci flat metrics
• Stability of symmetric spaces of noncompact type under Ricci flow