Paper · Aug 1, 2018

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

Stuart James Hall

math.DGarXiv:1808.00344v2
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Abstract

In this note we show that the bi-invariant Einstein metric on the compact Lie group is dynamically unstable as a fixed point of the Ricci flow. This completes the stability analysis for the bi-invariant metrics on the compact, connected, simple Lie groups. Interestingly, is the only unstable exceptional group.

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Submitted
Aug 1, 2018
Last revised
Dec 29, 2018 (v2)
Subjects
math.DG
DOI
10.1112/blms.12238

Comments: 7 pages; v2 typos fixed, rewritten following referee's suggestions, to appear in BLMS

Ricci flowEinstein metricsHomogeneous spacesStability & uniqueness

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)4Batat 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 On the linear stability of nearly-Kähler 6-manifolds (Changliang Wang and M. Y. -K. Wang, 2019)8Wang 2019 Ancient solutions of the homogeneous Ricci flow on flag manifolds (Stavros Anastassiou and Ioannis Chrysikos, 2020)3Anastassiou 2020 Rigidity of SUₙ-type symmetric spaces (Wafaa Batat, Stuart James Hall, Thomas Murphy et al., 2021)2Batat 2021 Classification of Left Invariant Riemannian metrics on Complex hyperbolic space (Andrijana Dekic, Marijana Babic and Srdjan Vukmirovic, 2021)5Dekic 2021 Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Carolyn Gordon and Michael Jablonski, 2021)6Gordon 2021 Positive Hermitian Curvature Flow on special linear groups and perfect solitons (James Stanfield, 2021)7Stanfield 2021
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 Einstein metrics, Stability & uniqueness · Same author Stuart James Hall · Shared terms “dynamically unstable”, “fixed point”

  2. Rigidity of -type symmetric spaces

    Wafaa Batat, Stuart James Hall, Thomas Murphy et al.

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

  3. Ancient solutions of the homogeneous Ricci flow on flag manifolds

    Stavros Anastassiou and Ioannis Chrysikos

    Topics Einstein metrics, Homogeneous spaces, Stability & uniqueness · Shared terms “invariant Einstein”, “simple Lie”, “fixed point”

  4. Destabilising compact warped product Einstein manifolds

    Wafaa Batat, Stuart James Hall and Thomas Murphy

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

  5. Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

    Andrijana Dekic, Marijana Babic and Srdjan Vukmirovic

    Topics Einstein metrics, Homogeneous spaces · Shared terms “Lie groups”

  6. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Topics Einstein metrics, Homogeneous spaces · Shared terms “Lie groups”

  7. Positive Hermitian Curvature Flow on special linear groups and perfect solitons

    James Stanfield

    Topics Homogeneous spaces, Stability & uniqueness · Shared terms “dynamically unstable”, “Lie groups”

  8. On the linear stability of nearly-Kähler -manifolds

    Changliang Wang and M. Y. -K. Wang

    Topics Stability & uniqueness · Shared terms “dynamically unstable”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.

  3. 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.