Eigenvalue Estimate for the basic Laplacian on manifolds with foliated boundary

In this paper, we give 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. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and the boundary is $\eta$-umbilical. This allows to characterize the quotient of $\mathbb{R}\times B^{\mathrm{\prime }}$ by some group $\mathrm{\Gamma }$ as being the limiting manifold. Here $B^{\mathrm{\prime }}$ 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.