Complete Minimal Surfaces in Rn

Let M be a compact oriented smooth manifold with boundary ∂M (possibly ∂M = O). Also let f: M → (N, ds2) be an immersion into a Riemannian manifold N. By a smooth variation of f we mean a smooth map F: I × M → N, I =(−1, 1) such that i) Ft = F(t, ·): M → N is an immersion; ii) F0 = f; iii) Ft | ∂M = f| ∂M for every t ∈ I.