Paper · Sep 22, 2020

Hopf type theorems for self-similar solutions of curvature flows in

Hilário Alencar, Gregório Silva Neto, Detang Zhou

math.DGWider flowsarXiv:2009.10494
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Abstract

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in . We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the -Gaussian curvature flow for . We remark that our result does not assume any embeddedness condition.

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Submitted
Sep 22, 2020
Subjects
math.DG, math.AP
Mean curvature flowOther curvature flowsGauss curvature flowCurvature pinchingSurfacesStability & uniqueness

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hopf type theorems in Riemannian manifolds

    Hilário Alencar, Gregório Silva Neto and Detang Zhou

    Topics Other curvature flows · Same authors Hilário Alencar, Gregório Silva Neto, Detang Zhou · Shared terms “hopf type”, “type theorem”

  2. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Surfaces, Stability & uniqueness · Same authors Hilário Alencar, Gregório Silva Neto, Detang Zhou

  3. On the spectrum and index of expanding and translating solitons of the mean curvature flow in

    Hilário Alencar and Gregório Silva NetoWider flows

    Topics Surfaces, Stability & uniqueness · Same authors Hilário Alencar, Gregório Silva Neto

  4. Poincaré type inequality for hypersurfaces and rigidity results

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows

    Topics Other curvature flows, Stability & uniqueness · Same authors Hilário Alencar, Gregório Silva Neto

  5. Gap theorems for complete self-shrinkers of -mean curvature flows

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Mean curvature flow · Same authors Hilário Alencar, Gregório Silva Neto

  6. Asymptotically conical shrinking ends of the -mean curvature flow: rotational construction and uniqueness

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows

    Topics Stability & uniqueness · Same authors Hilário Alencar, Gregório Silva Neto

  7. Half-space theorems for translating solitons of the r-mean curvature flow

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Mean curvature flow · Same authors Hilário Alencar, Gregório Silva Neto

  8. Self-similar solutions of -curvature flow

    Shanze Gao and Hui MaWider flows

    Topics Curvature pinching, Other curvature flows · Shared terms “round sphere”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.