Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

We consider the first eigenvalue λ1(Ω,σ) of the Laplacian with Robin boundary conditions on a compact Riemannian manifold Ω with smooth boundary, σ∈R being the Robin boundary parameter. When σ>0 we give a positive, sharp lower bound of λ1(Ω,σ) in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of Ω, a lower bound of the mean curvature of ∂Ω 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 λ1(Ω,σ) 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 λ1(Ω,σ) is uniformly bounded below by for all bounded domains in the hyperbolic space of dimension n, provided that the boundary parameter [Formula presented] (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed.