Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary, Part II

Abstract

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in El Chami et al. (Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, 2015) a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when $$p>1$$ . We then characterize the limiting case by showing that the manifold N is isometric to for some group $$\Gamma $$ where $$B'$$ denotes the unit closed ball. As a consequence, we describe the Riemannian product $${\mathbb {S}}^1\times {\mathbb {S}}^n$$ as the boundary of a manifold.