Scalar Curvature and Conformal Deformation of Hyperbolic Space

Let (Hm, gH), m ≥ 3, be the m-dimensional hyperbolic space with its Riemannian metric gH, of sectional curvature − 1; and let K be a smooth function on Hm. In the first part of this article we establish sufficient conditions for K to be-respectively, not to be-the scalar curvature of some complete metric u4/(m − 2)gH pointwise conformal to gH. In the second part we prove results for the two-dimensional case, singularity and uniquens questions.