Paper · Jan 7, 2021

A relative entropy and a unique continuation result for Ricci expanders

Alix Deruelle, Felix Schulze

math.DGarXiv:2101.02638
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Abstract

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two such self-similar solutions to the Ricci flow is well-defined.

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Submitted
Jan 7, 2021
Subjects
math.DG, math.AP

Comments: 63 pages

Ricci flowRicci solitonsWeak & singular flows

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 Weak stability of Ricci expanders with positive curvature operator (Alix Deruelle and Tobias Lamm, 2016)4Deruelle 2016 Expanding Kähler-Ricci solitons coming out of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2016)5Conlon 2016 Generic uniqueness of expanders with vanishing relative entropy (Alix Deruelle and Felix Schulze, 2018)1Deruelle 2018 On the regularity of Ricci flows coming out of metric spaces (Alix Deruelle, Felix Schulze and Miles Simon, 2019)2Deruelle 2019 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)3Deruelle 2022 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)6Deruelle 2024 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)8Deruelle 2025 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)7Deruelle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic uniqueness of expanders with vanishing relative entropy

    Alix Deruelle and Felix Schulze

    Topics Ricci solitons · Same authors Alix Deruelle, Felix Schulze · Shared terms “relative entropy”, “expanders”

  2. On the regularity of Ricci flows coming out of metric spaces

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Weak & singular flows · Same authors Alix Deruelle, Felix Schulze

  3. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Weak & singular flows · Same authors Alix Deruelle, Felix Schulze

  4. Weak stability of Ricci expanders with positive curvature operator

    Alix Deruelle and Tobias Lamm

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

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

    Ronan J. Conlon and Alix Deruelle

    Topics Weak & singular flows, Ricci solitons · Same author Alix Deruelle · Shared terms “solitons coming”, “expanding gradient”

  6. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Weak & singular flows · Same authors Alix Deruelle, Felix Schulze

  7. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al.

    Topics Ricci flow · Same authors Alix Deruelle, Felix Schulze

  8. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

    Topics Weak & singular flows · Same author Alix Deruelle

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.