The Yamabe problem and applications on noncompact complete Riemannian manifolds

We let (M, g) be a noncompact complete Riemannian manifold of dimension n greater than or equal to 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation -Delta au + [(n - 2)/(4(n - 1))]Su = qu((n+2)/(n-2)) on (M, g). This study finds restrictions on the existence of an injective conformal immersion of (M, g) into any compact Riemannian n-manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M, g) with some conditions at infinity.