Paper · Feb 25, 2016

Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds

Songbo Hou

math.DGarXiv:1602.07824v2
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Abstract

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.

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Submitted
Feb 25, 2016
Last revised
Aug 10, 2019 (v2)
Subjects
math.DG
Journal
Acta Math.Sin.(Engl.Ser.) 34 (2018), no. 7, 1179-1194

Comments: 20 pages

Ricci flow3-manifoldsHomogeneous spacesEigenvalues & 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 Ricci flow of model geometries (Songbo Hou, 2016)2Hou 2016 Eigenvalues of Weighted-Laplacian under the extended Ricci flow (Abimbola Abolarinwa, 2016)3Abolarinwa 2016 New Curvature Conditions for the Bochner Technique (Peter Petersen and Matthias Wink, 2019)7Petersen 2019 Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifolds (Songbo Hou and Shusen Yang, 2021)1Hou 2021 The curvature operator of the second kind in dimension three (Harry Fluck and Xiaolong Li, 2023)4Fluck 2023 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)5Colding 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)6Cunha 2023 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)8He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, 3-manifolds, Homogeneous spaces · Same author Songbo Hou · Shared terms “backward Ricci”, “locally homogeneous”, “Laplace”

  2. Eigenvalues under the Ricci flow of model geometries

    Songbo Hou

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

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

    Abimbola Abolarinwa

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

  4. The curvature operator of the second kind in dimension three

    Harry Fluck and Xiaolong Li

    Topics Eigenvalues & spectrum, 3-manifolds

  5. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum · Shared terms “lower bound”

  6. 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

  7. New Curvature Conditions for the Bochner Technique

    Peter Petersen and Matthias Wink

    Topics Eigenvalues & spectrum · Shared terms “lower bound”

  8. 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”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.