In this talk, I will describe a conformal technique inspired by the works of Fischer-Colbrie ('85) and Elbert–Nelli–Rosenberg ('07), and explain how it has recently been used successfully to obtain classification and rigidity results. In particular, I will discuss its application to an alternative proof of the stable Bernstein theorem in R^4 and to the study of noncompact critical metrics for the L^2-norm of the scalar curvature. The talk is based on joint works with G. Catino, D. D. Monticelli, and A. Roncoroni (Politecnico di Milano).