Paper · Apr 20, 2016

Eigenvalues of Weighted-Laplacian under the extended Ricci flow

Abimbola Abolarinwa

math.DGarXiv:1604.05884
View PDF View HTML arXiv page

As on arXiv

Abstract

Let be a symmetric diffusion operator with an invariant weighted volume measure on an -dimensional compact Riemannian manifold , where solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case . Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.

View PDF View HTML arXiv page
Submitted
Apr 20, 2016
Subjects
math.DG

Comments: 19 pages

Ricci flowSingularitiesEigenvalues & 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)1Hou 2016 Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifolds (Songbo Hou, 2016)4Hou 2016 First eigenvalues of geometric operators under the Yamabe flow (Pak Tung Ho, 2018)8Ho 2018 Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow (Shahroud Azami, 2019)7Azami 2019 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)3Colding 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 Evolution of Functionals Under Extended Ricci Flow (Shouvik Datta Choudhury, 2024)6Choudhury 2024 The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers (Junrong Yan, 2026)2Yan 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 Ricci flow of model geometries

    Songbo Hou

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

  2. The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers

    Junrong Yan

    Topics Eigenvalues & spectrum · Shared terms “weighted Laplacian”

  3. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum · Shared terms “weighted volume”, “Laplacian”

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

    Songbo Hou

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

  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 · Shared terms “compact Riemannian”, “Riemannian manifold”

  6. Evolution of Functionals Under Extended Ricci Flow

    Shouvik Datta Choudhury

    Topics Ricci flow · Shared terms “extended Ricci”, “compact Riemannian”, “Riemannian manifold”

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

    Shahroud AzamiWider flows

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

  8. First eigenvalues of geometric operators under the Yamabe flow

    Pak Tung HoWider flows

    Topics Eigenvalues & spectrum · Shared terms “nonzero eigenvalue”, “compact Riemannian”, “Laplacian”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.