Scalar curvature and conformal deformations of noncompact Riemannian manifolds (Q1365448)

The main aim of this paper is to study the following question. Problem. Let \(K\in C^\infty (M)\), where \((M,g)\) is an \(m\)-dimensional, connected, complete, noncompact Riemannian manifold, \(m\geq 3\). Does there exist a conformal deformation \(g_u\) \((g_u= u^{4/(m-2)}g\), \(u\) a smooth, positive function on \(M)\) of \(g\) such that \(K\) is the scalar curvature of \(g_u\) and \(g_u\) is complete? Essentially, the authors study in detail the case where \(K(x)\) is negative outside a compact set. They refine the results obtained by Avilès-McOwen, Jin, Gui-Wang and establish some uniqueness theorems as well.