Paper · Jun 10, 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, Marcus Antônio Mendonça Marrocos

math.DGarXiv:2306.06504v2
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Abstract

We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold . We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all –Riemannian metrics on or the set of all –diffeomorphisms on a domain into . In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase

Comments: Final version accepted for publication in Nonlinear Differential Equations and Applications (NoDEA)

Ricci flowEigenvalues & 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. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum

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

    Harry Fluck and Xiaolong Li PDF

    Topics Eigenvalues & spectrum

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

    Fei He PDF

    Topics Eigenvalues & spectrum

  4. A note on gradient Ricci soliton warped metrics

    José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro PDF

    Same authors José Nazareno Vieira Gomes, Marcus Antônio Mendonça Marrocos

  5. On a special class of gradient Ricci solitons

    José Nazareno Vieira Gomes and Marcus Antonio Mendonça Marrocos PDF

    Same authors José Nazareno Vieira Gomes, Marcus Antônio Mendonça Marrocos

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

    Junrong Yan PDF

    Topics Eigenvalues & spectrum

  7. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis PDF

    Topics Eigenvalues & spectrum

  8. 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

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.