Paper · Mar 30, 2023

The curvature operator of the second kind in dimension three

Harry Fluck, Xiaolong Li

math.DGarXiv:2303.17663v2
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Abstract

This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that -positive/-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all .

Comments: Comments are welcome. Fixed part 3 of theorem 3.1

Ricci flow3-manifoldsEigenvalues & spectrum

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 PDF

    Topics Eigenvalues & spectrum

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

    Fei He PDF

    Topics Eigenvalues & spectrum

  3. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum

  4. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee PDF

    Topics Eigenvalues & spectrum · Shared terms “curvature operator”

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

    Junrong Yan PDF

    Topics Eigenvalues & spectrum

  6. The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor

    Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos PDF

    Topics Eigenvalues & spectrum

  7. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

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

    Topics Eigenvalues & spectrum

  8. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al. PDF

    Topics 3-manifolds · Shared terms “curvature operator”

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) 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.