Eigenvalue Estimate for the basic Laplacian on manifolds with foliated boundary, part II

TL;DR

This paper extends lower bounds for the first eigenvalue of the basic Laplacian on manifolds with foliated boundary from 1-forms to p-forms, characterizing the limiting cases and specific manifolds.

Contribution

It generalizes previous results to p-forms and characterizes the limiting cases, including the Riemannian product of spheres as boundary.

Findings

Established sharp lower bounds for eigenvalues on basic p-forms.

Characterized the limiting cases as manifolds of the form R x B'/Γ.

Described the boundary as a Riemannian product of spheres.

Abstract

In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting case allows to characterize the manifold $\mathbb{R} \times B' / \Gamma$ for some group $\Gamma$, and where $B'$ denotes the unit closed ball. In particular, we describe the Riemannian product $\mathbb{S}^1\times \mathbb{S}^n$ as the boundary of a manifold.