First Robin Eigenvalue of the $p$-Laplacian on Riemannian Manifolds

We consider the first Robin eigenvalue $\l_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $\l_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $\l_p(M,\a)$ 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 $\a<0$. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\a \to +\infty$.