AUBIN’S CONTRIBUTION TO YAMABE PROBLEM 77 Aubin’s Contribution to Yamabe Problem

If (M, g) is a compact Riemannian manifold without boundary, of dimension n> 3, there is at least one metric g ′ conformal to g with constant scalar curvature. This statement, that H. Yamabe proved in [9], became a problem once it was detected a weak point in the original proof, as N.Trudinger remarked in [8]. The geometrical problem of finding a metric of the form g ′ = ϕ4/n−2g, with a smooth ϕ> 0 and with constant scalar curvature R ′, is expressed in terms of the existence of solutions of a given elliptic PDE in the manifold M. The equation relating R ′ with the scalar curvature of g, denoted R, via the Laplace operator of g i