CONFORMAL SCALAR CURVATURE EQUATIONS ON COMPLETE MANIFOLDS

This note contains considerations on the existence and non-existence p roblem of conformal scalar curvature equations on some complete manifo lds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radi us. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conforma lly compact manifolds. We obtain non-existence results on both types o f conditions. Then we study in more details the existence problem on m anifolds with cones, manifolds with cusps and conformally flat manifol ds of bounded positive scalar curvature.