Paper · Feb 19, 2024

Eigenvalue estimates on shrinkers

S. Brendle, R. Tsiamis

math.DGarXiv:2402.11803v3
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Abstract

We prove an eigenvalue estimate which holds on every properly embedded shrinker for mean curvature flow. This generalizes earlier work of Ding and Xin to the noncompact case.

Comments: Final version, to appear in Amer. J. Math

Mean curvature flowRicci solitonsBoundaries & non-compactEigenvalues & 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 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 Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)5Colding 2021 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)8Colding 2023 Rigidity of eigenvalues of shrinking Ricci solitons (Chang Li, Huaiyu Zhang and Xi Zhang, 2024)4Li 2024 An eigenvalue estimate for self-shrinkers in a Ricci shirinker (Franciele Conrado and Detang Zhou, 2025)1Conrado 2025 Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers (Xu Cheng, Franciele Conrado, Neilha Pinheiro et al., 2026)2Cheng 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)6Li 2026 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)7He 2026 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)3Brendle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An eigenvalue estimate for self-shrinkers in a Ricci shirinker

    Franciele Conrado and Detang Zhou PDF

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

  2. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

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

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

  3. Ricci flow on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis PDF

    Topics Boundaries & non-compact, Ricci solitons · Same authors S. Brendle, R. Tsiamis

  4. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang PDF

    Topics Eigenvalues & spectrum, Boundaries & non-compact, Ricci solitons

  5. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum, Boundaries & non-compact, Ricci solitons

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

    Fagui Li and Yuhang ZhaoWider flows PDF

    Topics Eigenvalues & spectrum · Shared terms “Ding”, “Xin”, “properly embedded”

  7. Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

    Fei He PDF

    Topics Eigenvalues & spectrum, Ricci solitons

  8. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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) 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.

  3. 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.