Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow
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Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow Stuart James Hall1 · Thomas Murphy2 · James Waldron1 Received: 9 December 2019 / Accepted: 19 September 2020 © Mathematica Josephina, Inc. 2020
Abstract Using a stability criterion due to Kröncke, we show, providing n = 2k, the Kähler–Einstein metric on the Grassmannian Grk (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 CPn 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. Keywords Symmetric space · Ricci flow · Stability of Einstein metrics
1 Introduction In 2013, Kröncke proved the surprising result that the Fubini–Study Kähler–Einstein metric on CPn , n > 1, is unstable as a fixed point of the Ricci flow [24]. More precisely, he showed that there are certain conformal (and hence non-Kähler) deformations of the Fubini–Study metric from which the Ricci flow never returns. This is in stark contrast to the behaviour of the Kähler–Ricci flow where Tian and Zhu [33] have shown that
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Thomas Murphy [email protected], [email protected] Stuart James Hall [email protected] James Waldron [email protected]
1
School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle-upon-Tyne NE1 7RU, England, UK
2
Department of Mathematics, California State University Fullerton, 800 N. State College Bld., Fullerton, CA 92831, USA
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S. J. Hall et al.
Kähler–Einstein metrics are essentially global attractors within their Kähler class. In [22] Knopf and Sesum give an independent verification of Kröncke’s result. The behaviour of Ricci flow on manifolds admitting Kähler metrics is a topic of current interest (see for example [15,16,20,26]). What Kröncke’s result suggests is that behaviour of the Ricci flow near the space of Kähler metrics is more complicated than was initially believed. If a Fano manifold M with Hodge number h 1,1 (M) > 1 admits a Kähler–Einstein metric then it can be destabilised by a harmonic perturbation within the Kähler cone. This method can be used to show many known examples of Kähler–Einstein metrics are unstable. However, as the complex dimension of the Fano manifold grows, there are numerous examples of Kähler–Einstein manifolds with h 1,1 (M) = 1. One such class of Kähler–Einstein manifolds are the compact, irreducible, Hermitian symmetric spaces. These manifolds were completely classified by Cartan into six types; there are four infinite families and two exceptional spaces. Each of these spaces admits a Kähler–Einstein metric unique up to automorphisms of the complex structure; this metric is the symmetric metric on each manifold. We will henceforth implicitly assume all manifo
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