Perturbative techniques in conformal geometry

Abstract

This paper reviews some results, mostly perturbative in nature, concerning the prescription of scalar curvature via conformal deformation of the metric on a compact manifold. This problem consists in solving a nonlinear elliptic equation with critical power, leading to non-compactness phenomena. We will see how some finite-dimensional reduction of the problem in certain situations, jointly with the variational structure, can help in finding existence and multiplicity results.