Fernando Codá Marques
Ph.D. (2003) Cornell University
Abstract: We prove that the set of solutions to the classical Yamabe equation, on a compact Riemannin n-manifold with positive Yamabe quotient, not necessarily locally conformally flat, is compact in the C^2 topology. Since we use the Positive Mass Theorem in the proof, we restrict ourselves to the cases 4 ≤ n ≤ 7. In the cases n = 6, 7, we also prove that the Weyl tensor has to vanish at a blowup point. The proofs are based on a careful blowup analysis of solutions. Given a compact n-manifold with umbilic boundary, n ≥ 9, finite Q (M, \partial M), such that the Weyl tensor does not vanish identically on \partial M, we show the existence of conformally related metrics with zero scalar curvature and constant mean curvature on \partial M. The proof of this result is based on an asymptotic analysis of the Sobolev quotients of explicitly defined test functions, using conformal Fermi coordinates.