Paper · Feb 15, 2016

Eigenvalues under the Ricci flow of model geometries

Songbo Hou

math.DGarXiv:1602.04597v2
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.

View PDF View HTML arXiv page
Submitted
Feb 15, 2016
Last revised
Aug 10, 2019 (v2)
Subjects
math.DG
Journal
Acta Math. Sinica (Chin. Ser.) 60 (2017), no. 4, 583-594

Comments: 15 pages

Ricci flowEigenvalues & 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 Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds (Songbo Hou, 2016)1Hou 2016 Eigenvalues of Weighted-Laplacian under the extended Ricci flow (Abimbola Abolarinwa, 2016)3Abolarinwa 2016 Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow (Shahroud Azami, 2019)7Azami 2019 Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds (Songbo Hou and Shusen Yang, 2021)2Hou 2021 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)4Colding 2023 Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications (Cleiton Lira Cunha, José Nazareno Vieira Gomes and Marcus Antônio Mendonça Marrocos, 2023)5Cunha 2023 Eigenvalue estimates on shrinkers (S. Brendle and R. Tsiamis, 2024)8Brendle 2024 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)6He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds

    Songbo Hou

    Topics Eigenvalues & spectrum · Same author Songbo Hou · Shared terms “evolving behaviors”, “Beltrami operator”, “Laplace Beltrami”

  2. Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds

    Songbo Hou and Shusen Yang

    Topics Eigenvalues & spectrum · Same author Songbo Hou · Shared terms “Laplace”, “Bianchi”, “lower bound”

  3. Eigenvalues of Weighted-Laplacian under the extended Ricci flow

    Abimbola Abolarinwa

    Topics Eigenvalues & spectrum · Shared terms “Beltrami operator”, “Laplace Beltrami”

  4. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum · Shared terms “lower bound”

  5. Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications

    Cleiton Lira Cunha, José Nazareno Vieira Gomes and Marcus Antônio Mendonça Marrocos

    Topics Eigenvalues & spectrum

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

    Fei He

    Topics Eigenvalues & spectrum · Shared terms “Laplace”, “lower bound”

  7. Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow

    Shahroud AzamiWider flows

    Topics Eigenvalues & spectrum · Shared terms “monotonic quantities”, “Laplace”

  8. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis

    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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.