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Sharp Spectral Gap Estimates on Manifolds under Integral Ricci Curvature Bounds

Xavier Ramos Olivé, Shoo Seto, Malik Tuerkoen

TL;DR

The paper proves sharp lower bounds for the first nonzero Laplacian eigenvalue $\lambda_1(M)$ on compact Riemannian manifolds under integral Ricci curvature bounds, quantified by $\overline{k}(p,K)$. It develops a gradient comparison framework using an auxiliary function $J$ to absorb the integral-curvature terms and a perturbed one-dimensional model to serve as a comparison device, ultimately relating $\lambda_1(M)$ to the one-dimensional model eigenvalue $\lambda_1(n,K,D)$. The results extend Kröger and Bakry–Qian to the integral-curvature setting and confirm a conjecture of Ramos et al. for $n\ge 3$, including an integral-curvature version of the Yang-type bound for $K<0$, with sharpness established in the limit of pointwise Ricci bounds. The approach yields a diameter-based reduction and yields a sharp spectral-gap estimate that is robust under small integral-curvature deviations, advancing spectral geometry under integral curvature constraints.

Abstract

We prove sharp spectral gap estimates on compact manifolds with integral curvature bounds. We generalize the results of Kröger (Kröger '92) as well as of Bakry and Qian (Bakry-Qian '00) to the case of integral curvature and confirm the conjecture in (Ramos et al. '20) for the case $n \geq 3$.

Sharp Spectral Gap Estimates on Manifolds under Integral Ricci Curvature Bounds

TL;DR

The paper proves sharp lower bounds for the first nonzero Laplacian eigenvalue on compact Riemannian manifolds under integral Ricci curvature bounds, quantified by . It develops a gradient comparison framework using an auxiliary function to absorb the integral-curvature terms and a perturbed one-dimensional model to serve as a comparison device, ultimately relating to the one-dimensional model eigenvalue . The results extend Kröger and Bakry–Qian to the integral-curvature setting and confirm a conjecture of Ramos et al. for , including an integral-curvature version of the Yang-type bound for , with sharpness established in the limit of pointwise Ricci bounds. The approach yields a diameter-based reduction and yields a sharp spectral-gap estimate that is robust under small integral-curvature deviations, advancing spectral geometry under integral curvature constraints.

Abstract

We prove sharp spectral gap estimates on compact manifolds with integral curvature bounds. We generalize the results of Kröger (Kröger '92) as well as of Bakry and Qian (Bakry-Qian '00) to the case of integral curvature and confirm the conjecture in (Ramos et al. '20) for the case .

Paper Structure

This paper contains 10 sections, 11 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Overview of the Paper
  3. Notation and Preliminaries
  4. Properties of the one-dimensional models
  5. The auxiliary function
  6. Gradient Comparison
  7. Maxima of Eigenfunctions
  8. Maximum Comparison
  9. Maximum Matching
  10. Proof of Theorem \ref{['thm: main theorem']}

Key Result

Theorem A

Let $M$ be an $n$-dimensional compact Riemannian manifold (with possibly non-empty convex and $C^2$ boundary), $\mathop{\mathrm{Ric}}\nolimits\geq (n-1)K$, $K\in \mathbb{R}$ and $\mathop{\mathrm{diam}}\nolimits(M)\leq D$. Let $\lambda_1(M)$ be the first nonzero eigenvalue of the Laplacian. Then where $\lambda_1(n,K,D)$ is the first nonzero eigenvalue of the one-dimensional eigenvalue problem whe

Theorems & Definitions (26)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: bakry2000some*Section 7, Theorem 13
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • ...and 16 more