The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

Abstract

We let (M,g) be a noncompact complete Riemannian manifold of dimension n ≥ 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 on −Δu+[(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.