Paper · Jan 27, 2016

Variation of Laplace spectra of compact "nearly" hyperbolic surfaces

Mayukh Mukherjee

math.DGarXiv:1601.07469
View PDF View HTML arXiv page

As on arXiv

Abstract

We use the time real analyticity of Ricci flow proved by Kotschwar to extend a result in , namely, we prove that the Laplace spectra of negatively curved compact surfaces having same genus , same area and same curvature bounds vary in a "controlled way", of which we give a quantitative estimate (Theorem 1.1 below). We also observe how said real analyticity can lead to unexpected conclusions about spectral properties of generic metrics on a compact surface of genus (Proposition 1.5 below).

View PDF View HTML arXiv page
Submitted
Jan 27, 2016
Subjects
math.DG

Comments: 7 pages, comments welcome!

Ricci flowEigenvalues & 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)4Abolarinwa 2016 On the spectra of geometric operators evolving with geometric flows (R. R. Mesquita and D. M. Tsonev, 2017)5Mesquita 2017 Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators (Sebastian Boldt and Batu Güneysu, 2020)8Boldt 2020 Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)6Colding 2021 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)3Cunha 2023 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)1He 2026 The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers (Junrong Yan, 2026)7Yan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

    Fei He

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

  2. Eigenvalues under the Ricci flow of model geometries

    Songbo Hou

    Topics Eigenvalues & spectrum · Shared terms “Laplace”

  3. 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 “variation”

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

    Abimbola Abolarinwa

    Topics Eigenvalues & spectrum · Shared terms “Laplace”

  5. On the spectra of geometric operators evolving with geometric flows

    R. R. Mesquita and D. M. TsonevWider flows

    Topics Eigenvalues & spectrum · Shared terms “variation”

  6. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi

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

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

    Junrong Yan

    Topics Eigenvalues & spectrum · Shared terms “spectral”

  8. Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

    Sebastian Boldt and Batu Güneysu

    Topics Eigenvalues & spectrum · Shared terms “spectral”

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.