The Zero Scalar Curvature Yamabe problem on noncompact manifolds with boundary

Abstract

Let (M n , g), n ≥ 3 be a noncompact complete Riemannian manifold with compact boundary and f a smooth function on ∂M. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of g that is complete, has zero scalar curvature on M, and has mean curvature f on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.