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

摘要

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 <= 7. We also show that, when n >= 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.