Paper · Jul 21, 2020

Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations

Peter Petersen, William Wylie

math.DGarXiv:2007.11058v2
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Abstract

We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, our results apply to homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics.

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Submitted
Jul 21, 2020
Last revised
Mar 18, 2022 (v2)
Subjects
math.DG

Comments: 33 pages, v2. Added a rigidity statement to Thm 1.5 in the Codazzi case, Added an appendix on related work in the Kaehler case, added a number of references

Other curvature flowsRicci solitonsEinstein metricsHomogeneous spacesStability & uniquenessRotational symmetry & warped products

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)1Batat 2016 Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)8Gomes 2017 Painlevé analysis of Ricci solitons over warped products (Alejandro Betancourt de la Parra, 2018)4de la Parra 2018 On G₂-structures, special metrics and related flows (Marisa Fernández, Anna Fino and Alberto Raffero, 2018)6Fernández 2018 On noncompact warped product Ricci solitons (Valter Borges, 2022)5Borges 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)7Kotschwar 2023 Hamilton's identity and rigidity of complete gradient solitons (Antonio W. Cunha, Antonio N. Silva and William Wylie, 2025)2Cunha 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)3Bhattacharya 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Destabilising compact warped product Einstein manifolds

    Wafaa Batat, Stuart James Hall and Thomas Murphy

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

  2. Hamilton's identity and rigidity of complete gradient solitons

    Antonio W. Cunha, Antonio N. Silva and William Wylie

    Topics Stability & uniqueness, Ricci solitons · Same author William Wylie · Shared terms “gradient solitons”, “gradient Ricci”

  3. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Rotational symmetry & warped products, Einstein metrics, Stability & uniqueness · Shared terms “gradient Ricci”

  4. Painlevé analysis of Ricci solitons over warped products

    Alejandro Betancourt de la Parra

    Topics Rotational symmetry & warped products, Einstein metrics, Homogeneous spaces

  5. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Einstein metrics, Stability & uniqueness · Shared terms “gradient Ricci”

  6. On -structures, special metrics and related flows

    Marisa Fernández, Anna Fino and Alberto Raffero

    Topics Einstein metrics, Homogeneous spaces, Ricci solitons

  7. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons

  8. Classification of base of warped product almost Ricci solitons

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

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons · Shared terms “Hessian”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) 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)

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

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

  3. 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.