A Lower Bound for the First Non-zero Basic Eigenvalue on a Singular Riemannian Foliation

TL;DR

This paper establishes lower bounds for the first non-zero basic eigenvalue on singular Riemannian foliations, generalizing classical estimates and proving a rigidity result characterizing cases of equality.

Contribution

It extends Zhong-Yang and Shi-Yang estimates to singular Riemannian foliations with basic mean curvature and proves a rigidity theorem for the eigenvalue equality case.

Findings

Derived lower bounds depending on Ricci curvature and leaf space diameter.

Generalized classical eigenvalue estimates for singular foliations.

Rigidity result characterizing the structure when bounds are attained.

Abstract

In this paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature of $M$ and the diameter of the leaf space $M/\mathcal{F}$. These can be regarded as generalized versions of the Zhong-Yang estimate and a generalized Shi-Yang's estimate for singular Riemannian foliations with basic mean curvature. We also provide a rigidity result corresponding to the generalized Zhong-Yang estimate, which is a generalized Hang-Wang rigidity for singular Riemannian foliations with basic mean curvature. More precisely, when the first basic eigenvalue $\lambda_1^B$ is equal to $\frac{\pi^2}{d_{M/\mathcal{F}^2}} $, where $d_{M/\mathcal{F}}$ is the diameter of the leaf space, $M$ is isometric to a mapping torus of an isometry $\varphi:N\to N$ where $N$ is an $(n-1)$-dimensional Riemannian manifold of nonnegative Ricci curvature and $\mathcal{F}$ has the form $\{[\{\text{point}\}\times N]\}$.