Some geometric and analytic aspects of the Yamabe Problem

We study the so-called "resolution criterion" for the Yamabe problem. After establishing what the yamabe constant is, we prove that every closed Riemannian manifold of dimension n>2 with Yamabe invariant strictly smaller than the one of the standard n-sphere admits a conformal change of metric that makes the scalar curvature constant. We do it by writing the problem as a PDE on a Riemannian manifold and solve it with a variational approach, properly adapted due to the criticality of our equation. In conclusion, we analyse deeply the main features of the problem on the standard n-sphere.