Paper · Sep 12, 2017

Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

Xu Cheng, Detang Zhou

math.DGWider flowsarXiv:1709.04086
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Abstract

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is the Euclidian subspace through the origin. Further, for self-expanders of codimension , we prove an inequality between the bottom of the spectrum of the drifted Laplacian and the bottom of the spectrum of weighted stability operator and that the hyperplane through the origin is the unique self-expander where the equality holds. Also we prove the uniqueness of hyperplane through the origin for mean convex self-expanders under some condition on the square of the norm of the second fundamental form.

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Submitted
Sep 12, 2017
Subjects
math.DG

Comments: 18 pages

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessEigenvalues & spectrum

Its neighbourhood

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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 Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons (Matheus Vieira and Detang Zhou, 2016)2Vieira 2016 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)8Bernstein 2019 Self-expanders of the mean curvature flow (Knut Smoczyk, 2020)6Smoczyk 2020 Volume properties and rigidity on self-expanders of mean curvature flow (Saul Ancari and Xu Cheng, 2020)1Ancari 2020 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)3Alencar 2022 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)7Leal 2024 An eigenvalue estimate for self-shrinkers in a Ricci shirinker (Franciele Conrado and Detang Zhou, 2025)5Conrado 2025 Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers (Xu Cheng, Franciele Conrado, Neilha Pinheiro et al., 2026)4Cheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume properties and rigidity on self-expanders of mean curvature flow

    Saul Ancari and Xu ChengWider flows

    Topics Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Same author Xu Cheng · Shared terms “self expanders”, “drifted Laplacian”, “stability operator”

  2. Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons

    Matheus Vieira and Detang Zhou

    Topics Eigenvalues & spectrum, Self-shrinkers & translators · Same author Detang Zhou · Shared terms “spectral properties”, “drifted Laplacian”

  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 Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “stability operator”

  4. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

    Xu Cheng, Franciele Conrado, Neilha Pinheiro et al.

    Topics Eigenvalues & spectrum · Same authors Xu Cheng, Detang Zhou · Shared terms “lower bound”, “equality”

  5. An eigenvalue estimate for self-shrinkers in a Ricci shirinker

    Franciele Conrado and Detang Zhou

    Topics Eigenvalues & spectrum, Self-shrinkers & translators · Same author Detang Zhou · Shared terms “drifted Laplacian”, “lower bound”, “Next”

  6. Self-expanders of the mean curvature flow

    Knut SmoczykWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanding”, “self expanders”, “expanding solutions”

  7. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou

    Topics Eigenvalues & spectrum · Same author Detang Zhou · Shared terms “expanders”, “Laplacian”

  8. Topological Uniqueness for Self-expanders of Small Entropy

    Jacob Bernstein and Lu WangWider flows

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

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) 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) 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) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 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.