Complete conformal metrics with prescribed scalar curvature on subdomains of a compact manifold

Source abstract

Let (M, g) be a Riemannian manifold of dimension n ≥ 3 and ĝ another metric on M which is pointwise conformai to g . It can be written where u is a positive smooth function on M . Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equation where ▽u denotes the Laplacian of u , defined with respect to the metric g .