A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Abstract

Let (M, g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n = 6, 7, 8, provided that the Weyl tensor is always not vanishing on the boundary.