Solução do problema de Yamabe

Abstract

In this work we expose the complete solution of the Yamabe Problem: “Given a compact connected Riemannian manifold (M; g) of dimension n ≥ 3; find a metric conformal to g with constant scalar curvature”. Initially, Yamabe transforms this problem to a problem of elliptic operators. Together with the works of Trundiger and Aubin, we show that for a certain class of manifolds there is always solution. We will also see with results due to Aubin, simplifed by the use of conformal normal coordinates, that many of the manifolds belong to this class. An important case is the solution on the Sphere, in which we mainly use Yamabe’s .Teorem and Obata’s .Teorem. Finally, we show explain how Schoen uses the Positive Mass .Teorem to complete the results obtained by Aubin and conclude that the problem always has a solution.