  We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply
connected manifolds of constant curvature (negative or positive) that imply the
earlier ones for Euclidean space of Kenig, Ruiz and the second author
\cite{KRS}. In the case of the sphere we take advantage of the fact that the
half-wave group of the natural shifted Laplacian is periodic. In the case of
hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas
restriction theorem.
