A priori estimates for the Yamabe problem in the non-locally conformally flat case
Fernando C Marques
Given a compact Riemannian manifold $(M^n,g)$ 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 where the Positive Mass Theorem is known to be true, that is, when $n \leq 7$. We also show that, when $n \geq 6$, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.