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

1. Problem Statement

The paper addresses the spectral geometry problem of estimating the first non-zero basic eigenvalue ($\lambda_1^B$) of the Laplacian on a singular Riemannian foliation $(M, \mathcal{F})$.

• Context: Classical Riemannian geometry has established lower bounds for the first eigenvalue $\lambda_1$ of a compact manifold $M$ based on Ricci curvature and diameter (e.g., Lichnerowicz, Zhong-Yang, Shi-Zhang).
• Specific Challenge: In the context of singular Riemannian foliations, one studies functions that are constant on the leaves (basic functions). The goal is to generalize classical eigenvalue estimates to this setting.
• Key Difficulty: While Lichnerowicz-type estimates (dependent only on curvature) can be trivially adapted, estimates involving the diameter are non-trivial. The paper seeks to replace the diameter of the ambient manifold ($d_M$) with the diameter of the leaf space ($d_{M/\mathcal{F}}$) in these estimates, assuming the foliation admits a basic mean curvature.

2. Methodology

The author employs a combination of geometric analysis, gradient estimates, and comparison theorems adapted to the leaf space structure.

• Basic Mean Curvature Assumption: The paper assumes the foliation admits a basic mean curvature. This is a crucial condition because it ensures that the ordinary Laplacian $\Delta$ coincides with the basic Laplacian $\Delta_B$ when acting on basic functions. This allows the use of standard analytic tools on the ambient manifold $M$ while respecting the foliation structure.
• Leaf Space Geometry: The leaf space $M/\mathcal{F}$ is treated as a metric space (specifically, a manifold submetry). The diameter $d_{M/\mathcal{F}}$ is defined as the supremum of distances between leaves.
• Gradient Estimates (Zhong-Yang Generalization):
  • The proof adapts the Zhong-Yang gradient estimate technique.
  • Instead of integrating along geodesics in $M$, the author integrates along geodesics connecting leaves (specifically, the shortest geodesic between a point on a level set of the eigenfunction and the level set of the opposite sign).
  • By utilizing the global equidistance property of closed leaves in singular Riemannian foliations, the length of these geodesics is bounded by $d_{M/\mathcal{F}}$ rather than $d_M$.
• Rigidity Analysis:
  • The paper analyzes the equality case of the lower bound.
  • It utilizes and modifies recent results by G. Xu and X. Xue regarding the splitting of manifolds with non-negative Ricci curvature when the gradient of the eigenfunction achieves a specific bound.
  • The author corrects a previous claim in the literature regarding the structure of such manifolds, proving they are mapping tori rather than simple products.
• Comparison with Model Spaces (Shi-Zhang Generalization):
  • For positive Ricci curvature, the paper uses a 1D Sturm-Liouville comparison argument (following Bakry-Qian and Shi-Zhang).
  • It compares the basic eigenvalue to the first non-zero Neumann eigenvalue of a specific 1D operator on an interval of length $d_{M/\mathcal{F}}$.

3. Key Contributions and Results

A. Generalized Zhong-Yang Estimate (Non-negative Ricci Curvature)

Theorem 1.1: Let $(M, g)$ be a compact Riemannian manifold with $\text{Ric} \ge 0$, and $\mathcal{F}$ a singular Riemannian foliation with closed leaves and basic mean curvature. Then:
$$\lambda_1^B \ge \frac{\pi^2}{d_{M/\mathcal{F}}^2}$$

• Significance: This generalizes the optimal Zhong-Yang estimate ($\lambda_1 \ge \pi^2/d_M^2$) to the basic spectrum, replacing the ambient diameter with the leaf space diameter.

B. Rigidity Result (Generalized Hang-Wang)

Theorem 1.2: If equality holds ($\lambda_1^B = \pi^2/d_{M/\mathcal{F}}^2$), then:

1. $M$ is isometric to a mapping torus $R \times_{\mathbb{Z}} N$, where $N$ is a compact $(n-1)$-manifold with $\text{Ric} \ge 0$.
2. The action of $\mathbb{Z}$ is defined by an isometry $\phi: N \to N$.
3. The foliation $\mathcal{F}$ consists of the fibers $\{ \{t\} \times N \}$, making $M/\mathcal{F}$ isometric to a circle of radius $1/\sqrt{\lambda_1^B}$.

• Correction: The author notes that previous literature suggested $M$ must be a product $S^1 \times N$. This paper proves that $M$ can be a non-trivial mapping torus (e.g., Klein bottle or Seifert manifolds), providing a more precise structural characterization.

C. Generalized Shi-Zhang Estimate (Positive Ricci Curvature)

Theorem 1.3: If $\text{Ric} \ge (n-1)K \ge 0$, then for any $s \in (0, 1)$:
$$\lambda_1^B \ge 4s(1-s)\frac{\pi^2}{d_{M/\mathcal{F}}^2} + s(n-1)K$$

• Optimization: The paper analyzes the optimal choice of $s$. For small diameters, the optimal $s$ yields a sharper bound than the standard Li-type estimate:
  $$\lambda_1^B \ge \frac{\pi^2}{d_{M/\mathcal{F}}^2} + \frac{1}{2}(n-1)K + \frac{(n-1)^2 K^2 d_{M/\mathcal{F}}^2}{16\pi^2}$$

4. Significance and Implications

1. Unification of Spectral Geometry: The results unify classical eigenvalue estimates (Zhong-Yang, Shi-Zhang) with the theory of singular Riemannian foliations. They demonstrate that the spectral properties of the basic Laplacian are governed by the geometry of the leaf space rather than the ambient manifold's full geometry.
2. Rigidity of Structure: The rigidity theorem provides a deep topological and geometric classification of manifolds where the basic eigenvalue achieves its theoretical minimum. It reveals that such manifolds are essentially "twisted" products (mapping tori) over a circle, generalizing the Hang-Wang result for spheres and circles.
3. Optimality: The paper argues that these estimates are sharp. In the case where the foliation consists of points (trivial foliation), the results recover the classical theorems. In the case of Riemannian submersions with minimal fibers, the basic spectrum coincides with the spectrum of the base, validating the use of $d_{M/\mathcal{F}}$.
4. Counter-Examples to Naive Generalizations: The paper highlights that one cannot simply apply classical rigidity theorems to the base manifold of a submersion to deduce the structure of the total space $M$. The structure of the foliation itself (specifically, whether it is a product or a mapping torus) is essential.

5. Conclusion

Bach Tran successfully extends fundamental eigenvalue estimates to the setting of singular Riemannian foliations with basic mean curvature. By rigorously adapting gradient estimates and splitting theorems to the leaf space metric, the paper establishes that the first non-zero basic eigenvalue is bounded below by a function of the leaf space diameter and Ricci curvature, with equality implying a specific mapping torus structure. This work bridges the gap between global Riemannian geometry and the spectral theory of foliated spaces.