In the recent past a lot of authors have devoted systematic efforts to indagate problems related to compactness and degeneration phenomenons in classes of surfaces with integral bounds on curvature, both on the front of geometric measure theory and on that of differential geometry. As a general reference reviewing the main results obtained in this area by differential geometers, we refer the reader to  while, unfortunately, we have no paper so efficaciously summarizing the theory developed in the setting of geometric measure theory to recommend. These two points of view are deeply different and only a few attempts to bridge the gap existing between them have been done up to now. As very interesting examples of steps in this direction from geometric measure theory, we have to mention the papers about curvature varifolds by Hutchinson (, , , ) and Mantegazza () where a notion of generalized second fundamental form is defined and applied to variational problems. We also have to recall the papers on generalized Gauss graphs (, , , , , , , , ). In particular, in paper , special generalized Gauss graphs have been used to prove a result of differential geometry ([14, Theorem 7.1]) which generalizes a theorem by Langer .