On the fundamental tone of the $p$-Laplacian on Riemannian manifolds and applications

We present a general lower bound for the fundamental tone for the $p$-Laplacian on Riemannian manifolds carrying a special kind of function. We then apply our result to the cases of negatively curved simply connected manifolds, a class of warped product manifolds and for a class of Riemannian submersions.