Paper · Mar 21, 2016

The index of shrinkers of the mean curvature flow

Zihan Hans Liu

math.DGarXiv:1603.06539
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As on arXiv

Abstract

We introduce a notion of index for shrinkers of the mean curvature flow. We then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in R^3, namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the plane. We also provide a generalization of the result to higher dimensions.

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Submitted
Mar 21, 2016
Subjects
math.DG
Mean curvature flowRicci solitonsRotational 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 Type-II singularities of two-convex immersed mean curvature flow (Theodora Bourni and Mat Langford, 2016)6Bourni 2016 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)8Brendle 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)7Li 2018 Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori (Yakov Berchenko-Kogan, 2020)1Berchenko-Ko 2020 Numerically computing the index of mean curvature flow self-shrinkers (Yakov Berchenko-Kogan, 2020)3Berchenko-Ko 2020 A new proof for the existence of rotationally symmetric gradient Ricci solitons (Shu-Yu Hsu, 2021)4Hsu 2021 Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions (Kin Ming Hui, 2021)2Hui 2021 Instability of mean curvature flow in expander coordinates (Ben S. Lambert and Oliver C. Schnürer, 2026)5Lambert 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

    Yakov Berchenko-KoganWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “higher dimensions”

  2. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “higher dimensions”, “rotationally symmetric”

  3. Numerically computing the index of mean curvature flow self-shrinkers

    Yakov Berchenko-KoganWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  4. A new proof for the existence of rotationally symmetric gradient Ricci solitons

    Shu-Yu Hsu

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “rotationally symmetric”

  5. Instability of mean curvature flow in expander coordinates

    Ben S. Lambert and Oliver C. Schnürer

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “rotationally symmetric”

  6. Type-II singularities of two-convex immersed mean curvature flow

    Theodora Bourni and Mat LangfordWider flows

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  7. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “higher dimensions”, “rotationally symmetric”

  8. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products · Shared terms “higher dimensions”, “rotationally symmetric”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.