Blow-Up Phenomena for a Boundary Yamabe Problem with Umbilic Boundary

Source abstract

We study a linear boundary perturbation of the problem of prescribing the scalar curvature K and the boundary mean curvature H on a Riemannian manifold with umbilic boundary. Assuming that the Weyl tensor is nowhere vanishing along the boundary, we treat separately the cases in which K and H are constant and non-constant. In the negative scalar-curvature regime K<0, and in dimensions n≥8, we prove the existence of positive solutions that blow up at the boundary as the perturbation parameter tends to zero.