Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary

Abstract

On a compact Riemannian manifold whose boundary is endowed with a Riemannian flow, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic 1-forms. The equality case gives rise to a particular geometry of the flow and of the boundary. Namely, we prove that the flow is a local product and the boundary is $$\eta $$ -umbilical. This allows to characterize the quotient of $${\mathbb {R}}\times B'$$ by some group $$\Gamma $$ as the limiting manifold. Here $$B'$$ denotes the unit closed ball. Finally, we deduce several rigidity results describing the product $${\mathbb {S}}^1\times {\mathbb {S}}^n$$ as the boundary of a manifold.