The Yamabe constant on noncompact manifolds

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe constant at infinity is even locally constant with respect to this topology. We also discuss to which extent the Yamabe constant is continuous with respect to coarser topologies on the space of Riemannian metrics.