On the Yamabe Problem concerning the compact locally conformally flat manifolds

Abstract

For all known locally conformally flat compact Riemannian manifolds ( M n , g) ( n > 2), with infinite fundamental group, we give the complete proof of Aubin's conjecture on scalar curvature. That solves the Yamabe Problem for these manifolds. There exists a metric g′ conformal to g, such that vol g′ = 1 and whose scalar curvature R′ is constant and satisfies R′ < n(n − 1) ω n 2 n , where ω n is the volume of the sphere S n with radius 1.