Paper · Jul 31, 2023

New expanding Ricci solitons starting in dimension four

Jan Nienhaus, Matthias Wink

math.DGarXiv:2307.16683
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Abstract

We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants.

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Submitted
Jul 31, 2023
Subjects
math.DG
Journal
J. Geom. Anal. 34, 327 (2024)
DOI
10.1007/s12220-024-01778-4

Comments: 15 pages

Ricci solitonsEinstein metricsRicci-flat & Calabi–Yau4-manifoldsScalar curvatureBoundaries & non-compact

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 2021, 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 Gradient Ricci solitons carrying a closed conformal vector field (J. F. Siva Filho and R. Sharma, 2021)4Filho 2021 Curvature estimates for four-dimensional complete gradient expanding Ricci solitons (Huai-Dong Cao and Tianbo Liu, 2021)1Cao 2021 Rigidity results for shrinking and expanding Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)2Leandro 2022 Classification of Gradient Ricci solitons with harmonic Weyl curvature (Jongsu Kim, 2023)6Kim 2023 Geometry and Analysis of Gradient Ricci Solitons in Dimension Four (Xiaodong Cao and Hung Tran, 2024)8Cao 2024 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)7Lafuente 2025 Curvature pinching of asymptotically conical gradient expanding Ricci solitons (Huai-Dong Cao and Junming Xie, 2025)5Cao 2025 A note on the classification of four-dimensional gradient steady and expanding Ricci solitons (Huai-Dong Cao and Junming Xie, 2026)3Cao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”

  2. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”, “Einstein”

  3. A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

    Huai-Dong Cao and Junming Xie

    Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Shared terms “expanding Ricci”, “gradient expanding”, “asymptotically conical”

  4. Gradient Ricci solitons carrying a closed conformal vector field

    J. F. Siva Filho and R. Sharma

    Topics Ricci-flat & Calabi–Yau, 4-manifolds, Einstein metrics

  5. Curvature pinching of asymptotically conical gradient expanding Ricci solitons

    Huai-Dong Cao and Junming Xie

    Topics 4-manifolds, Scalar curvature, Boundaries & non-compact · Shared terms “expanding Ricci”, “gradient expanding”, “asymptotically conical”

  6. Classification of Gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Ricci-flat & Calabi–Yau, 4-manifolds, Einstein metrics

  7. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson

    Topics 4-manifolds, Ricci solitons · Shared terms “expanding Ricci”, “vector bundles”, “Einstein”

  8. Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

    Xiaodong Cao and Hung Tran

    Topics 4-manifolds, Ricci solitons · Shared terms “dimension four”

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 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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