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

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 $λ_1^B$ is equal to $\frac{π^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 $\{[\{\text{point}\}\times N]\}$.

Figures (2)

• Figure 1: The picture illustrates how the geodesic $\gamma_y$, from $y=\gamma_y(0)\in P$, traverses along all nodal domains and turn back to $P$ the second time at $\varphi(y)=\gamma_y\left(\frac{\mathcal{N}(u)}{\sqrt{\lambda}}\right)$.
• Figure 2: To show $L_1=L_2$, the idea is to prove that if $u,v$ are points that realize the distances from $s$ to each leaf $L_1,L_2$ respectively, then $u$ and $v$ must be the same point. In the picture, we see that if $L_1\neq L_2$, then $u\neq v$.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Example 1
• Example 2
• Proposition 4.1
• proof
• proof : Proof of Theorem \ref{['main1']}
• Proposition 4.2: Proposition 3.2 (N) in xuxue