Paper · Jul 19, 2016

Destabilising compact warped product Einstein manifolds

Wafaa Batat, Stuart James Hall, Thomas Murphy

math.DGarXiv:1607.05766v3
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Abstract

The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics and Ricci solitons, we introduce a new destabilising perturbation (the Ricci variation) and show that certain infinite families of warped product Einstein metrics will be unstable in high dimensions.

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Submitted
Jul 19, 2016
Last revised
Jan 8, 2019 (v3)
Subjects
math.DG, math-ph

Comments: V3: Final Version. To appear in Comm. Anal. Geom

Ricci flowRicci solitonsEinstein metricsAlmost & η-Ricci solitonsStability & uniquenessRotational symmetry & warped products

Its neighbourhood

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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 Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)4Gomes 2017 Ricci Almost Solitons on semi-Riemannian Warped Products (Keti Tenenblat and Valter Borges, 2017)7Tenenblat 2017 Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow (Stuart James Hall, Thomas Murphy and James Waldron, 2018)3Hall 2018 The canonical Einstein metric on G₂ is dynamically unstable under the Ricci flow (Stuart James Hall, 2018)8Hall 2018 Rigidity of SUₙ-type symmetric spaces (Wafaa Batat, Stuart James Hall, Thomas Murphy et al., 2021)2Batat 2021 On noncompact warped product Ricci solitons (Valter Borges, 2022)5Borges 2022 A Characterization of Quasi-Einstein Metrics (Antonio Airton Freitas Filho, 2025)6Filho 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)1Bhattacharya 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”, “fixed point”

  2. Rigidity of -type symmetric spaces

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

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

  3. 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 authors Stuart James Hall, Thomas Murphy · Shared terms “unstable”, “fixed point”, “generalises”

  4. Classification of base of warped product almost Ricci solitons

    José N. V. Gomes and Manoel V. M. Neto

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”

  5. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Einstein metrics, Stability & uniqueness

  6. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”

  7. Ricci Almost Solitons on semi-Riemannian Warped Products

    Keti Tenenblat and Valter Borges

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics

  8. 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 “unstable”, “fixed point”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.