Blow-up phenomena for scalar-flat metrics on manifolds with boundary

Let (Mn,g) be a compact Riemannian manifold with boundary ∂M. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have ∂M as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball Bn, in dimensions n⩾25, for which this set is non-compact. These manifolds have umbilic boundary, but they are not conformally equivalent to Bn.