Paper · Aug 26, 2019

New Curvature Conditions for the Bochner Technique

Peter Petersen, Matthias Wink

math.DGarXiv:1908.09958v4
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Abstract

We prove a vanishing and estimation theorem for the -Betti number of closed -dimensional Riemannian manifolds with a lower bound on the average of the lowest eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions we obtain vanishing of the Betti numbers provided that the curvature operator is -positive. As Böhm-Wilking observed, -positivity of the curvature operator is not preserved by the Ricci flow.

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Submitted
Aug 26, 2019
Last revised
Sep 29, 2020 (v4)
Subjects
math.DG
Journal
Invent. Math. 224, 33-54 (2021)
DOI
10.1007/s00222-020-01003-3

Comments: An edited version to appear in Invent. Math. Corollary 3.3 and Proposition 3.5 to be published elsewhere

Ricci flowEigenvalues & spectrum

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    Harry Fluck and Xiaolong Li

    Topics Eigenvalues & spectrum · Shared terms “curvature operator”, “preserved”

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

    Fei He

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

  3. The Ricci flow under almost non-negative curvature conditions

    Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking

    Topics Eigenvalues & spectrum · Shared terms “curvature condition”, “curvature operator”

  4. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi

    Topics Eigenvalues & spectrum · Shared terms “lower bound”

  5. Eigenvalues under the Ricci flow of model geometries

    Songbo Hou

    Topics Eigenvalues & spectrum · Shared terms “lower bound”

  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

    Topics Eigenvalues & spectrum · Shared terms “Riemannian manifold”

  7. Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

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

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

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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.