Scalar curvatures on noncompact Riemann manifolds (Q1842119)

Let \((M,g)\) be a complete noncompact simply connected Riemannian manifold with scalar curvature \(K\), \(\dim M \geq 3\). The author studies the problem of finding a complete conformal metric \(\widetilde {g}\) on \(M\) such that its scalar curvature equals a prescribed function \(\widetilde {K}\). The case \(\widetilde {K} = \) const is known as the Yamabe problem. The author gives conditions under which existence of such a metric \(\widetilde {g}\) is guaranteed. For example, if \(K \leq 0\), \(\widetilde {K} < 0\), and outside a compact set \(K/ \widetilde {K} \geq \varepsilon > 0\), then there exists a conformal complete metric \(\widetilde {g}\) with scalar curvature \(\widetilde {K}\). Moreover, a class of manifolds is described for which the problem under consideration is not solvable.