Paper · Sep 12, 2016

Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons

Matheus Vieira, Detang Zhou

math.DGarXiv:1609.03247
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Abstract

In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.

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Submitted
Sep 12, 2016
Subjects
math.DG

Comments: 19 pages

Ricci solitonsSelf-shrinkers & translatorsEigenvalues & 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 Spectral properties and rigidity for self-expanding solutions of the mean curvature flows (Xu Cheng and Detang Zhou, 2017)1Cheng 2017 Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature (Xu Cheng, Matheus Vieira and Detang Zhou, 2019)4Cheng 2019 Stability properties of complete self-shrinking surfaces in ℝ³ (Hilário Alencar, Gregório Silva Neto and Detang Zhou, 2021)7Alencar 2021 Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons (Igor Miranda and Matheus Vieira, 2021)6Miranda 2021 Rigidity of eigenvalues of shrinking Ricci solitons (Chang Li, Huaiyu Zhang and Xi Zhang, 2024)8Li 2024 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)3Leal 2024 An eigenvalue estimate for self-shrinkers in a Ricci shirinker (Franciele Conrado and Detang Zhou, 2025)2Conrado 2025 Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers (Xu Cheng, Franciele Conrado, Neilha Pinheiro et al., 2026)5Cheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Detang Zhou · Shared terms “spectral properties”, “drifted Laplacian”

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

    Franciele Conrado and Detang Zhou

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

  3. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou

    Topics Eigenvalues & spectrum, Ricci solitons · Same authors Matheus Vieira, Detang Zhou · Shared terms “Laplacian”

  4. Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature

    Xu Cheng, Matheus Vieira and Detang Zhou

    Topics Ricci solitons · Same authors Matheus Vieira, Detang Zhou · Shared terms “properly immersed”

  5. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

    Xu Cheng, Franciele Conrado, Neilha Pinheiro et al.

    Topics Eigenvalues & spectrum, Ricci solitons · Same author Detang Zhou

  6. Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons

    Igor Miranda and Matheus Vieira

    Topics Ricci solitons · Same author Matheus Vieira · Shared terms “cylinder shrinking”, “shrinking Ricci”, “ambient space”

  7. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Self-shrinkers & translators · Same author Detang Zhou

  8. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “shrinking Ricci”, “drifted Laplacian”

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.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.