Paper · Dec 15, 2016

Volume growth, entropy and stability for translating solitons

Qiang Guang

math.DGWider flowsarXiv:1612.05312
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Abstract

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give a curvature estimate for translators in with small entropy. Finally, we estimate the spectrum of the stability operator for translators and give a rigidity result of -stable translators.

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Submitted
Dec 15, 2016
Subjects
math.DG
Journal
Comm. Anal. Geom. 27 (2019), no. 1, 47-72
DOI
10.4310/CAG.2019.v27.n1.a2

Comments: 20 pages, accepted by Comm. Anal. Geom

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessEigenvalues & spectrum

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 On the rigidity of mean convex self-shrinkers (Qiang Guang and Jonathan J. Zhu, 2016)6Guang 2016 Spectral properties and rigidity for self-expanding solutions of the mean curvature flows (Xu Cheng and Detang Zhou, 2017)5Cheng 2017 Volume properties and rigidity on self-expanders of mean curvature flow (Saul Ancari and Xu Cheng, 2020)2Ancari 2020 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)7Zhu 2020 Eigenvalue Inequalities for the Clamped Plate Problem of L²_ν Operator (Lingzhong Zeng, 2021)4Zeng 2021 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)1Alencar 2022 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)8White 2022 Rigidity and Non-existence Results of λ-Translating Solitons (Li Xiang and Sun Jun, 2026)3Xiang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “translating solitons”, “volume growth”, “stability operator”

  2. 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 · Shared terms “volume growth”, “stability operator”, “properly”

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

    Li Xiang and Sun JunWider flows

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

  4. Eigenvalue Inequalities for the Clamped Plate Problem of Operator

    Lingzhong ZengWider flows

    Topics Eigenvalues & spectrum, Self-shrinkers & translators · Shared terms “translating solitons”

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

    Xu Cheng and Detang ZhouWider flows

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

  6. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Qiang Guang · Shared terms “curvature estimates”

  7. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators · Shared terms “translating solitons”

  8. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows

    Topics Self-shrinkers & translators · Shared terms “Huisken monotonicity”, “monotonicity formula”

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.