Paper · Feb 14, 2021

Rigidity of -type symmetric spaces

Wafaa Batat, Stuart James Hall, Thomas Murphy, James Waldron

math.DGarXiv:2102.07168v2
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Abstract

We prove that the bi-invariant Einstein metric on is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-Kähler, non-product example of this phenomenon adding to the famous example of found by Koiso. We apply our methods to derive similar solitonic rigidity results for the Kähler–Einstein metrics on `odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.

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Submitted
Feb 14, 2021
Last revised
Nov 20, 2021 (v2)
Subjects
math.DG

Comments: 28 pages; v2 typos and small errors fixed

Ricci flowEinstein metricsKähler geometryStability & uniquenessKähler–Einstein & K-stability

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Destabilising compact warped product Einstein manifolds (Wafaa Batat, Stuart James Hall and Thomas Murphy, 2016)2Batat 2016 Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow (Stuart James Hall, Thomas Murphy and James Waldron, 2018)1Hall 2018 The canonical Einstein metric on G₂ is dynamically unstable under the Ricci flow (Stuart James Hall, 2018)3Hall 2018 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)6Liu 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)8Lee 2019 Examples of K-unstable Fano manifolds (Thibaut Delcroix, 2019)7Delcroix 2019 3-symmetric spaces, Ricci solitons, and homogeneous structures (Thomas Murphy and Paul-Andi Nagy, 2025)4Murphy 2025 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)5Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

    Stuart James Hall, Thomas Murphy and James Waldron

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same authors Stuart James Hall, Thomas Murphy, James Waldron · Shared terms “symmetric spaces”, “Grassmannian”, “dynamical”

  2. Destabilising compact warped product Einstein manifolds

    Wafaa Batat, Stuart James Hall and Thomas Murphy

    Topics Einstein metrics, Stability & uniqueness · Same authors Wafaa Batat, Stuart James Hall, Thomas Murphy

  3. The canonical Einstein metric on is dynamically unstable under the Ricci flow

    Stuart James Hall

    Topics Einstein metrics, Stability & uniqueness · Same author Stuart James Hall · Shared terms “invariant Einstein”

  4. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy

    Topics Kähler geometry · Same author Thomas Murphy · Shared terms “symmetric spaces”, “moduli space”

  5. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  6. Stability of the conical Kähler-Ricci flows on Fano manifolds

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  7. Examples of K-unstable Fano manifolds

    Thibaut Delcroix

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  8. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

Read around it

Background reading

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

  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.