Compactness for conformal scalar-flat metrics on umbilic boundary manifolds

Let (M, g) a compact Riemannian n-dimensional manifold with umbilic boundary. It is well known that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have partial derivative M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set, provided n = 8 and the Weyl tensor of the boundary is always different from zero, or if n > 8 and the Weyl tensor of M is always different from zero on the boundary. (C) 2020 Elsevier Ltd. All rights reserved.