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 complete manifold. This solution helps to obtain a general result on the prescribed scalar curvature problem, in the positive case [11]. This paper deals with the study of the prescribed curvature problem on noncompact complete manifolds. On compact manifolds it has been solved under conditions on the function f . The main difficulties in this case is the concentration phenomenon [19]. On a complete manifold having positive Yamabe invariant, another phenomena can appear. Conditions must be prescribed on the function f in order to avoid it. Some results concerning the existence of conformal metrics of negative scalar curvature manifolds were obtained by Aviles-MacOwen [5]. They proved that it is possible to solve the Yamabe equation with a constant value equal to -1 for the prescribed curvature . In the positive case, Kim [14] solved the Yamabe problem on a complete manifold assuming that the Yamabe quotient is less than a certain number. (Vn, g) will denote a complete Riemannian manifold of dimension greater or equal to 3 and R(g), the scalar curvature. We use the notation N = 2n n−2 , R̄ = R cn where cn = 4(n−1) n−2 . We shall assume that the conformal Yamabe invariant is positive : μ = inf H1(Vn) ∫ Vn (∇u)2 + R̄u2 ( ∫ Vn uN )2/N > 0 (1) There are many instances of such manifolds : for example asymptotically flat manifolds with positive scalar curvature. We can construct other examples