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 $\{[\{\text{point}\}\times N]\}$.
The study of how curvature influences the eigenvalues of the Laplacian has been a central theme in Riemannian geometry since the 1950s. One interesting problem in that trend is giving estimates for lower and upper bounds of the first eigenvalue Date: February 20, 2026.
of the Laplacian of a compact Riemannian manifold under curvature assumptions (in this paper, specifically, lower bounds on the Ricci curvature).
Recall that the Laplacian, or the Laplace-Beltrami operator on a Riemannian manifold (M n , g) is given by ∆ = div(∇).
A value λ will be called an eigenvalue of the Laplacian of M if there exists a non-zero function f ∈ C ∞ (M ) (that we call an eigenfunction with the eigenvalue λ) such that ∆f = -λf.
When M is compact, we know the fact that λ is non-negative.
Let λ 1 (M ) be the first non-zero eigenvalue of the Laplacian on M and λ 1 if we are clear about which manifold we work on.
In the case that Ric(M ) ≥ (n -1)K ≥ 0 for some constant K, many results have been found: Lichnerowicz [Lic58] and Obata [Oba62] proved that on a closed Riemannian manifold with K > 0, one has λ 1 ≥ nK and the equality holds if and only if M is isometric to the n-sphere of radius 1/ √ K. In the case Ric(M ) ≥ 0, we cannot get a positive lower bound without a restriction of the diameter d M of M : for example, in the case of torus, if we increase the diameter arbitrarily and approaching infinity, then the first eigenvalue approaches 0. Moreover, the diameter restriction is automatic in the case of K > 0 by Bonnet-Myer’s theorem. Hence, the diameter restriction is necessary. By Li and Yau [LY80] and the improvement by Li [Li79], we get a gradient estimate for the first nontrivial eigenfunction and derive that
Zhong and Yang [ZY84] enhanced this result to be an optimal one for the case of Ric(M ) ≥ 0:
Hang and Wang [HW07] later provided and proved the rigidity statemtent for Zhong-Yang estimate: the equality holds if and only if M is isometric to a circle of radius d M /π. We can see there are many works about general optimal lower bound estimate for λ 1 for a given lower bound (n -1)K of the Ricci curvature, for example, in [Kro92; CW97; AC13; ZW17], and see also [BQ00;AN12;He13;Oli+20]. Inspired by foundational results of Lichnerowicz and Zhong-Yang, Li had conjectured that the first positive eigenvalue should satisfy the following estimate (see [Yan99])
The conjecture greatly motivates many related studies in this area and an effort to prove the so-called Li’s conjecture will unify Zhong-Yang’s estimate and Lichnerowicz’s estimate. Many people tried to prove this conjecture, and in particular, some towards improved inequalities of the form
for some constant α were made. A remarkable result by Shi and Zhang in [SZ07] proved that for any s ∈ (0, 1)
Notice that for s = 1 2 , we have a particular estimate
In [AC13], B. Andrews and J. Clutterbuck showed that the inequality of this form 1.1 with α = 1 2 is the best possible constant and this means Li’s conjecture is false. For more detail of the history, see [He13].
So how is the lower bound estimates of the first eigenvalue related to the object we will discuss in this paper, the first basic eigenvalue on a singular Riemannian foliation? We first introduce the concept of singular Riemannian foliations. A singular Riemannian foliation on a Riemannian manifold M , roughly speaking, is a partition of M into connected injectively immersed submanifolds (we call leaves), which are locally equidistant to one another (see subsection 2.1). One class of examples of singular Riemannian foliations are orbits of isometric actions of Riemannian manifolds, that we call homogeneous. Another class of examples are connected components of the fibers of Riemannian submersions, that we call simple. For more detail about homogeneous and simple singular Riemannian foliations, see [Rad].
Given (M, F) a singular Riemannian foliation, we define F-basic function, is a function that is constant on each leaf of F. When the know which F we talk about, we can call it briefly basic function. This allows us to define F -basic Laplacian as an operator that sends basic smooth functions to basic smooth functions and define Fbasic eigenvalues, that we will discuss more detail in Section 2. When (M, F ) admits a basic mean curvature, it turns out that the ordinary Laplacian coincides with the basic Laplacian, and any results that work for the ordinary Laplacian can be used to derive corresponding results for the basic Laplacian. In the case of estimate a lower bound for the first non-zero basic eigenvalue λ B 1 , if we use Lichnerowiz’s estimate, we can get λ B 1 ≥ λ 1 ≥ nK but it does not imply any better result because nK does not depend on F that we consider, and we do not see the difference between Lichnerowicz’s estimate for the first non-zero eigenvalue and Lichnerowic’z estimate for the first non-zero basic eigenvalue. However, for the estimates involving the diameter like in [ZY84; Kro92; CW
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