Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

We consider the first eigenvalue lambda(1)(Omega, sigma) of the Laplacian with Robin boundary conditions on a compact Riemannian manifold Omega with smooth boundary, sigma is an element of R being the Robin boundary parameter. When sigma > 0 we give a positive, sharp lower bound of lambda(1)(Omega, sigma) in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of Omega, a lower bound of the mean curvature of partial derivative Omega and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for lambda(1)(Omega, sigma) obtained in Euclidean space by Giorgi and Smits [10], to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that lambda(1)(Omega, sigma) is uniformly bounded below by (n-1)(2)/4 for all bounded domains in the hyperbolic space of dimension n, provided that the boundary parameter sigma >= n-1/2 (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed. (C) 2019 Elsevier Inc. All rights reserved.