Paper · Nov 11, 2022

Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen, Guofang Wei

math.DGarXiv:2211.06403v2
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Abstract

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in and . Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.

Comments: 14 pages. To appear in Communications in Analysis and Geometry

Ricci flowSurfacesEigenvalues & spectrum

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  1. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero PDF

    Topics Surfaces · Shared terms “positive curvature”

  2. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “Laplacian”

  3. Variations of the harmonic entropy and linear drift on negatively curved surfaces

    Karen Butt, Alena Erchenko and Tristan Humbert PDF

    Topics Surfaces · Shared terms “curved surfaces”

  4. Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and D Ricci Flows

    Mohammadjavad Habibivostakolaei PDF

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “Dirichlet”

  5. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

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

  6. Contractive transport maps from to nearly spherical surfaces with positive Ricci curvature

    Jordan Serres PDF

    Topics Surfaces · Shared terms “curved surfaces”, “positively curved”

  7. Optimal growth bounds for eigenfunctions

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum · Shared terms “eigenfunction”, “previously”, “Laplacian”

  8. Monotonicity of the Liouville entropy along the Ricci flow on surfaces

    Karen Butt, Alena Erchenko, Tristan Humbert et al. PDF

    Topics Surfaces · Shared terms “lower bound”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.