Paper · May 3, 2023

Eigenvalue lower bounds and splitting for modified Ricci flow

Tobias Holck Colding, William P. Minicozzi

math.DGarXiv:2305.02416
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Abstract

We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.

Ricci flowEigenvalues & spectrum

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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 2021, 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 Optimal growth bounds for eigenfunctions (Tobias Holck Colding and William P. Minicozzi, 2021)1Colding 2021 Singularities of Ricci flow and diffeomorphisms (Tobias Holck Colding and William P. Minicozzi, 2021)3Colding 2021 Singularities and diffeomorphisms (Tobias Holck Colding and William P. Minicozzi, 2022)8Colding 2022 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)6Cunha 2023 Rigidity of eigenvalues of shrinking Ricci solitons (Chang Li, Huaiyu Zhang and Xi Zhang, 2024)4Li 2024 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)5Leal 2024 A Characterization of Quasi-Einstein Metrics (Antonio Airton Freitas Filho, 2025)2Filho 2025 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)7He 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “drift Laplacian”, “splitting”

  2. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho PDF

    Topics Eigenvalues & spectrum · Shared terms “modified Ricci”

  3. Singularities of Ricci flow and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Ricci flow · Same authors Tobias Holck Colding, William P. Minicozzi · Shared terms “splitting”

  4. Rigidity of eigenvalues of shrinking Ricci solitons

    Chang Li, Huaiyu Zhang and Xi Zhang PDF

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

  5. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum · Shared terms “drift Laplacian”

  6. 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

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

    Fei He PDF

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

  8. Singularities and diffeomorphisms

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Ricci flow · Same authors Tobias Holck Colding, William P. Minicozzi

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.