Paper · Oct 9, 2024

Spectrum of the drift Laplacian on Ricci expanders

Helton Leal, Matheus Vieira, Detang Zhou

math.DGarXiv:2410.07372
PDF Abstract page

As on arXiv

Abstract

In this paper, we study the spectrum of the drift Laplacian on Ricci expanders. We show that the spectrum is discrete when the potential function is proper, and we show that the hypothesis on the properness of the potential function cannot be removed. We also extend previous results concerning the asymptotic behavior of the potential function on Ricci expanders. This allows us to conclude that the drift Laplacian has discrete spectrum on Ricci expanders whose Ricci curvature is bounded below by a suitable constant, possibly negative. Further, we compute all the eigenvalues of the drift Laplacian on rigid expanders and rigid shrinkers. Lastly, we investigate the second eigenvalue of the drift Laplacian on rigid Ricci expanders whose Einstein factor is a closed hyperbolic Riemann surface.

Comments: 17 pages

Ricci solitonsSurfacesEigenvalues & 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 On the spectrum and index of expanding and translating solitons of the mean curvature flow in ℝ³ (Hilário Alencar and Gregório Silva Neto, 2022)4Alencar 2022 Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature (Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al., 2022)6Khan 2022 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)3Colding 2023 Rigidity of eigenvalues of shrinking Ricci solitons (Chang Li, Huaiyu Zhang and Xi Zhang, 2024)5Li 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 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)7He 2026 Variations of the harmonic entropy and linear drift on negatively curved surfaces (Karen Butt, Alena Erchenko and Tristan Humbert, 2026)8Butt 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 · Same author Detang Zhou · Shared terms “potential function”, “Laplacian”, “Einstein”

  2. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

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

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

  3. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum · Shared terms “drift Laplacian”

  4. On the spectrum and index of expanding and translating solitons of the mean curvature flow in

    Hilário Alencar and Gregório Silva NetoWider flows PDF

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “expander”

  5. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “discrete spectrum”, “Laplacian”, “Lastly”

  6. Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

    Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al. PDF

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “Laplacian”

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

    Fei He PDF

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

  8. Variations of the harmonic entropy and linear drift on negatively curved surfaces

    Karen Butt, Alena Erchenko and Tristan Humbert PDF

    Topics Surfaces

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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