Paper · Mar 11, 2022

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

Hilário Alencar, Gregório Silva Neto

math.DGWider flowsarXiv:2203.06214v2
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Abstract

In this paper we prove that two-dimensional translating solitons in with finite -index are homeomorphic to a plane or a cylinder and that a two-dimensional self-expander with finite -index and sub exponential weighted volume growth has finite topology. We also prove that translating solitons and self-expanders have finite topology, provided the bottom of the spectrum of the -stability operator is bounded from below and their weighted volume have subexponential growth.

Comments: We made a mistake in the proof of Lemma 2.5, we thanks PR. Márcio Batista (UFAL) for pointed us out this

Mean curvature flowSelf-shrinkers & translatorsSurfacesStability & uniquenessEigenvalues & spectrum

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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 2021, 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 Poincaré type inequality for hypersurfaces and rigidity results (Hilário Alencar, Márcio Batista and Gregório Silva Neto, 2022)8Alencar 2022 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)5Neto 2023 Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow (Debora Impera and Michele Rimoldi, 2023)6Impera 2023 Gap theorems for complete self-shrinkers of r-mean curvature flows (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2024)3Alencar 2024 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)7Leal 2024 Half-space theorems for translating solitons of the r-mean curvature flow (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2026)1Alencar 2026 Rigidity and Non-existence Results of λ-Translating Solitons (Li Xiang and Sun Jun, 2026)4Xiang 2026 Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness (Hilário Alencar, Márcio Batista and Gregório Silva Neto, 2026)2Alencar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Half-space theorems for translating solitons of the r-mean curvature flow

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

    Topics Self-shrinkers & translators · Same authors Hilário Alencar, Gregório Silva Neto · Shared terms “translating solitons”

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

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows PDF

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

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

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

    Topics Self-shrinkers & translators · Same authors Hilário Alencar, Gregório Silva Neto

  4. Rigidity and Non-existence Results of -Translating Solitons

    Li Xiang and Sun JunWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “volume growth”, “exponential”

  5. Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Self-shrinkers & translators · Same author Gregório Silva Neto · Shared terms “translating solitons”

  6. Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow

    Debora Impera and Michele RimoldiWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “weighted volume”

  7. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “expander”

  8. Poincaré type inequality for hypersurfaces and rigidity results

    Hilário Alencar, Márcio Batista and Gregório Silva NetoWider flows PDF

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

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  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.