First Robin eigenvalue of the p-Laplacian on Riemannian manifolds

We consider the first Robin eigenvalue lambda(p)(M, alpha) for the p-Laplacian on a compact Riemannian manifold M with nonempty smooth boundary, with alpha is an element of R being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for lambda(p)( M, a). Secondly, when alpha > 0 we establish sharp lower bound of lambda p(M, alpha) in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when alpha < 0. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the p-Laplacian when letting alpha -> +infinity.