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

We consider the first Robin eigenvalue ${\text{l}}_p(M,\text{a})$ 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 ${\text{l}}_p(M,\text{a})$. Secondly, when $\text{a}>0$ we establish sharp lower bound of ${\text{l}}_p(M,\text{a})$ 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 $\text{a}<0$. Our results cover corresponding comparison theorems for the first Dirichlet eigenvalue of the $p$-Laplacian when letting $\text{a}\to +\mathrm{\infty }$.