Prescribed scalar curvature problem on complete manifolds

Conditions on the geometric structure of a complete Riemannian manifold are given to solve the prescribed scalar curvature problem in the positive case. The conformal metric obtained is complete. A minimizing sequence is constructed which converges strongly to a solution. In a second part, the prescribed scalar curvature problem of zero value is solved which is equivalent to find a solution to a linear PDE on a complete manifold. This solution helps to obtain a general result on the prescribed scalar curvature problem, in the positive case. (C) 2001 Editions scientifiques et medicales Elsevier SAS.