A priori estimates for the Yamabe problem in the non-locally conformally flat case

Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show that, when the dimension is greater than or equal to 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.