Paper · Aug 23, 2017

Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

Jacob Bernstein

math.DGWider flowsarXiv:1708.07085v2
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Abstract

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone. Another consequence is a unique continuation property for self-expanders of the mean curvature flow that flow from a cone.

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Submitted
Aug 23, 2017
Last revised
Dec 13, 2017 (v2)
Subjects
math.DG, math.AP

Comments: 26 pages. Fixed typos and included additional references. Weakened hypotheses of Theorem 9.1

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessBoundaries & non-compactEigenvalues & 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 Spectral properties and rigidity for self-expanding solutions of the mean curvature flows (Xu Cheng and Detang Zhou, 2017)7Cheng 2017 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)2Bernstein 2017 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)3Bernstein 2018 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)5Bernstein 2019 A mountain-pass theorem for asymptotically conical self-expanders (Jacob Bernstein and Lu Wang, 2020)4Bernstein 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)8Khan 2020 Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)6Colding 2021 Asymptotically conical shrinking ends of the r-mean curvature flow: rotational construction and uniqueness (Hilário Alencar, Márcio Batista and Gregório Silva Neto, 2026)1Alencar 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotically conical shrinking ends of the -mean curvature flow: rotational construction and uniqueness

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    Topics Eigenvalues & spectrum, Boundaries & non-compact, Stability & uniqueness · Shared terms “almost eigenfunction”, “drift Laplacian”, “conical ends”

  2. The space of asymptotically conical self-expanders of mean curvature flow

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    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Jacob Bernstein · Shared terms “self expanders”, “asymptotically conical”

  3. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

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    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Jacob Bernstein · Shared terms “self expanders”, “asymptotically conical”

  4. A mountain-pass theorem for asymptotically conical self-expanders

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Jacob Bernstein · Shared terms “self expanders”, “asymptotically conical”

  5. Topological Uniqueness for Self-expanders of Small Entropy

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Jacob Bernstein · Shared terms “self expanders”

  6. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum, Boundaries & non-compact · Shared terms “eigenfunctions”, “drift Laplacian”, “Gaussian weight”

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

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

  8. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

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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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.