Paper · Jan 3, 2016

Weak stability of Ricci expanders with positive curvature operator

Alix Deruelle, Tobias Lamm

math.DGarXiv:1601.00325
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Abstract

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

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Submitted
Jan 3, 2016
Subjects
math.DG

Comments: 37 pages

Ricci solitonsSphere theoremsStability & 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 Manifolds with positive curvature operators are space forms (Christoph Boehm and Burkhard Wilking, 2006)1Boehm 2006 Expanding Kähler-Ricci solitons coming out of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2016)2Conlon 2016 A relative entropy and a unique continuation result for Ricci expanders (Alix Deruelle and Felix Schulze, 2021)4Deruelle 2021 Ricci flow and PIC1 (Peter M. Topping, 2023)5Topping 2023 A family of Kähler flying wing steady Ricci solitons (Pak-Yeung Chan, Ronan J. Conlon and Yi Lai, 2024)8Chan 2024 Orbifold singularity formation along ancient and immortal Ricci flows (Alix Deruelle and Tristan Ozuch, 2024)6Deruelle 2024 Rigidity of positively curved Steady gradient Ricci solitons on orbifolds (Yuxing Deng, 2025)7Deng 2025 Steady Gradient Ricci Solitons with O(p)× O(q) Symmetry (Lucas Lavoyer and Luke T. Peachey, 2025)3Lavoyer 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Manifolds with positive curvature operators are space forms

    Christoph Boehm and Burkhard WilkingThe canon

    Topics Sphere theorems · Shared terms “positive curvature”, “curvature operator”

  2. Expanding Kähler-Ricci solitons coming out of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Sphere theorems, Stability & uniqueness, Ricci solitons · Same author Alix Deruelle · Shared terms “positive curvature”, “expanding gradient”, “curvature operator”

  3. Steady Gradient Ricci Solitons with Symmetry

    Lucas Lavoyer and Luke T. Peachey

    Topics Sphere theorems, Ricci solitons · Shared terms “positive curvature”, “curvature operator”, “gradient Ricci”

  4. A relative entropy and a unique continuation result for Ricci expanders

    Alix Deruelle and Felix Schulze

    Topics Ricci solitons · Same author Alix Deruelle · Shared terms “Ricci expander”, “expanding gradient”, “gradient Ricci”

  5. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Shared terms “positive curvature”

  6. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch

    Topics Sphere theorems, Stability & uniqueness, Ricci solitons · Same author Alix Deruelle

  7. Rigidity of positively curved Steady gradient Ricci solitons on orbifolds

    Yuxing Deng

    Topics Sphere theorems, Stability & uniqueness, Ricci solitons · Shared terms “positive curvature”, “curvature operator”, “curvature decay”

  8. A family of Kähler flying wing steady Ricci solitons

    Pak-Yeung Chan, Ronan J. Conlon and Yi Lai

    Topics Sphere theorems, Stability & uniqueness, Ricci solitons · Shared terms “positive curvature”, “curvature operator”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  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.