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

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 $\mathbb{R}\times B^{\prime }/\Gamma$ for some group $\Gamma$, and where $B^{\prime }$ 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.