On surfaces of finite total curvature

We consider surfaces M immersed into Rⁿand we prove that the quantity ∫_M|A|²(where A is the second fundamental form) controls in many ways the behaviour of conformal parametrizations of M. If M is complete, connected, noncompact and ∫_M|A|²< ∞ we obtain a more or less complete picture of the behaviour of the immersions. In particular we prove that under these assumptions the immersions are proper. Moreover, if ∫_M|A|²≤ 4π or if n = 3 and ∫_M|A|²< 8π, then M is embedded. We also prove that conformal parametrizations of graphs of W^{2, 2}functions on R²exist, are bilipschitz and the conformal metric is continuous. The paper was inspired by recent results of T.Toro.