Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary

Abstract

Let (Mn,g) be a compact manifold with boundary, with finite Sobolev quotient Q(Mn, ∂M). We prove that there exists a conformal deformation which is scalar-flat and has constant boundary mean curvature, if n = 4 or 5 and the boundary is not umbilic. In particular, we prove such existence for any smooth and bounded open set of the Euclidean space, finishing the remaining cases of a theorem of J.F. Escobar.