Compactness and non-compactness for the Yamabe problem on manifolds with boundary

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension n <= 24. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counterexamples to compactness when n >= 25. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which will be fundamental for a study of the non-umbilic case.