Existence and compactness theorems for the Yamabe problem on manifolds with boundary
Manifold with boundary | Yamabe problem | scalar curvature | Mean curvature | Weyl tensor | trace-free second fundamental form.
Let (M,g) be a compact Riemannian manifold with boundary. This work addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. In the first part we prove an existence theorem, for the umbilic boundary case, that finishes some remaining cases of this problem. In the second part we prove that the whole set of solutions to this problem is compact for dimensions n>=7 under the generic condition that the boundary trace-free second fundamental form is nonzero everywhere.