Paper · Jan 29, 2021

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

Songbo Hou, Shusen Yang

math.DGarXiv:2101.12464
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Abstract

Let be the first eigenvalue of under the backward Ricci flow on locally homogeneous 3-manifolds, where is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of . In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, approaches zero, where .

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Submitted
Jan 29, 2021
Subjects
math.DG

Comments: 15 pages

Ricci flow3-manifoldsScalar curvatureHomogeneous spacesEigenvalues & spectrum

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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 Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (Fangcheng Guo, Guanghan Li and Chuanxi Wu, 2016)4Guo 2016 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)7Abolarinwa 2016 The curvature operator of the second kind in dimension three (Harry Fluck and Xiaolong Li, 2023)3Fluck 2023 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)8Colding 2023 Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers (Xu Cheng, Franciele Conrado, Neilha Pinheiro et al., 2026)5Cheng 2026 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, 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 “Laplace”, “Bianchi”, “lower bound”

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

    Harry Fluck and Xiaolong Li

    Topics Eigenvalues & spectrum, 3-manifolds

  4. Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

    Fangcheng Guo, Guanghan Li and Chuanxi WuWider flows

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

  5. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

    Xu Cheng, Franciele Conrado, Neilha Pinheiro et al.

    Topics Eigenvalues & spectrum, Scalar curvature · Shared terms “lower bound”, “Let”

  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. Eigenvalues of Weighted-Laplacian under the extended Ricci flow

    Abimbola Abolarinwa

    Topics Eigenvalues & spectrum · Shared terms “Laplace”, “Let”

  8. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum · Shared terms “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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

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.