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

Stuart James Hall; Thomas Murphy; James Waldron

Paper · Jun 5, 2018

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

Stuart James Hall, Thomas Murphy, James Waldron

math.DGarXiv:1806.01722v3

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Abstract

Using a stability criterion due to Kröncke, we show, providing $n \neq = 2 k$ , 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 $S U (n)$ . 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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Submitted: Jun 5, 2018
Last revised: Sep 20, 2020 (v3)
Subjects: math.DG

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

Ricci flow Einstein metrics Kähler geometry Stability & uniqueness Kähler–Einstein & K-stability

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

[1] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[2] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[3] 11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

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

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