Paper · May 6, 2025

An eigenvalue estimate for self-shrinkers in a Ricci shirinker

Franciele Conrado, Detang Zhou

math.DGarXiv:2505.03499
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Abstract

In this paper, we study the drifted Laplacian on a hypersurface in a Ricci shrinker . We prove that the spectrum of is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of when the hypersurface is an embedded -minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.

Comments: 19 pages. Comments are welcome

Ricci solitonsEinstein metricsSelf-shrinkers & translatorsEigenvalues & spectrumMinimal & CMC surfaces

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 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 A Minkowski type inequality for manifolds with positive spectrum (Ovidiu Munteanu and Jiaping Wang, 2023)6Munteanu 2023 The Wasserstein distance for Ricci shrinkers (Franciele Conrado and Detang Zhou, 2023)4Conrado 2023 Eigenvalue estimates on shrinkers (S. Brendle and R. Tsiamis, 2024)2Brendle 2024 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 Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers (Xu Cheng, Franciele Conrado, Neilha Pinheiro et al., 2026)1Cheng 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)5Li 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)7Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

    Xu Cheng, Franciele Conrado, Neilha Pinheiro et al. PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Same authors Franciele Conrado, Detang Zhou · Shared terms “eigenvalue estimate”, “Ricci shrinkers”, “lower bound”

  2. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “eigenvalue estimate”, “Ding”, “Xin”

  3. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Same author Detang Zhou · Shared terms “potential function”, “Laplacian”, “Einstein”

  4. The Wasserstein distance for Ricci shrinkers

    Franciele Conrado and Detang Zhou PDF

    Topics Ricci solitons · Same authors Franciele Conrado, Detang Zhou · Shared terms “Ricci shrinkers”

  5. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows PDF

    Topics Eigenvalues & spectrum, Self-shrinkers & translators · Shared terms “Ding”, “Xin”, “Tsiamis”

  6. A Minkowski type inequality for manifolds with positive spectrum

    Ovidiu Munteanu and Jiaping Wang PDF

    Topics Minimal & CMC surfaces, Eigenvalues & spectrum, Ricci solitons · Shared terms “minimal hypersurface”, “compact minimal”, “lower bound”

  7. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al. PDF

    Topics Einstein metrics, Ricci solitons · Same author Detang Zhou · Shared terms “Ricci shrinkers”

  8. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang PDF

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

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.