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

Using a stability criterion due to Kröncke, we show, providing n ≠ 2 k , the Kähler–Einstein metric on the Grassmannian G r k ( 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 Krön...

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Bibliographic Details
Published in:The Journal of geometric analysis Vol. 31; no. 6; pp. 6195 - 6218
Main Authors: Hall, Stuart James, Murphy, Thomas, Waldron, James
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2021
Springer Nature B.V
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamic stability
Dynamical Systems and Ergodic Theory
Flow stability
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Orbits
Stability criteria
Ricci flow
Symmetric space
Stability of Einstein metrics
ISSN:1050-6926, 1559-002X
Online Access:Get full text
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Summary:Using a stability criterion due to Kröncke, we show, providing n ≠ 2 k , the Kähler–Einstein metric on the Grassmannian G r k ( 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 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.
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-020-00524-w