On the lowest eigenvalue of the Hodge Laplacian on compact,negatively curved domains

We study the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute or relative boundary conditions. We prove a series of lower bounds when the domain is starlike or p-convex and the ambient manifold has pinched negative curvature. The bounds are sharp for starlike domains. We then compute the asymptotics of the first eigenvalue of hyperbolic balls of large radius. Finally, we give lower bounds also for Euclidean domains.