Paper · Dec 20, 2018

Generic uniqueness of expanders with vanishing relative entropy

Alix Deruelle, Felix Schulze

math.DGarXiv:1812.08504v2
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Abstract

We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.

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Submitted
Dec 20, 2018
Last revised
Apr 23, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 31 pages. Final version, to appear in Math. Annalen

Mean curvature flowRicci solitonsStability & 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 Weak stability of Ricci expanders with positive curvature operator (Alix Deruelle and Tobias Lamm, 2016)4Deruelle 2016 On the regularity of Ricci flows coming out of metric spaces (Alix Deruelle, Felix Schulze and Miles Simon, 2019)6Deruelle 2019 Relative expander entropy in the presence of a two-sided obstacle and applications (Jacob Bernstein and Lu Wang, 2019)7Bernstein 2019 Mean curvature flow with generic initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2020)3Chodosh 2020 A relative entropy and a unique continuation result for Ricci expanders (Alix Deruelle and Felix Schulze, 2021)1Deruelle 2021 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)2Deruelle 2022 Relative Entropy of Hypersurfaces in Hyperbolic Space (Junfu Yao, 2022)8Yao 2022 On the Hamilton-Lott conjecture in higher dimensions (Alix Deruelle, Felix Schulze and Miles Simon, 2024)5Deruelle 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A relative entropy and a unique continuation result for Ricci expanders

    Alix Deruelle and Felix Schulze

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

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

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Stability & uniqueness · Same authors Alix Deruelle, Felix Schulze

  3. Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.

    Topics Stability & uniqueness, Ricci solitons · Same author Felix Schulze

  4. Weak stability of Ricci expanders with positive curvature operator

    Alix Deruelle and Tobias Lamm

    Topics Stability & uniqueness, Ricci solitons · Same author Alix Deruelle · Shared terms “expanders”

  5. On the Hamilton-Lott conjecture in higher dimensions

    Alix Deruelle, Felix Schulze and Miles Simon

    Same authors Alix Deruelle, Felix Schulze

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

    Alix Deruelle, Felix Schulze and Miles Simon

    Same authors Alix Deruelle, Felix Schulze

  7. Relative expander entropy in the presence of a two-sided obstacle and applications

    Jacob Bernstein and Lu Wang

    Topics Ricci solitons · Shared terms “relative entropy”, “expanders”

  8. Relative Entropy of Hypersurfaces in Hyperbolic Space

    Junfu YaoWider flows

    Topics Mean curvature flow · Shared terms “relative entropy”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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. 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.