Abstract: Given an integral M-currrent T₀ in R^m+k and a tensor H of type(m,l)on R^m+k with values orthogonal to each of its arguments we proved in a previous paper [3] the existence of an integral m-current T=γ(M,θ.ζ)with boundary ∂T₀ and mean curvature vector H by minimizing an appropriate functional on suitable subclasses of the set of all integral currents.In this paper we discuss the existence and structure of oriented tangent cones C of T at points x∈spt(T)-spt(∂T),especially we show that C is locally mass minimizing.

Key words: integral currents, generalized mean curvature, tangent cones