Minimizing properties of networks via global and local calibrations

In this note, we prove that minimal networks enjoy minimizing properties for the length functional. A minimal network is, roughly speaking, a subset of (Formula presented.) composed of straight segments joining at triple junctions forming angles equal to (Formula presented.); in particular such objects are just critical points of the length functional a priori. We show that a minimal network (Formula presented.) : (i) minimizes mass among currents with coefficients in an explicit group (independent of (Formula presented.)) having the same boundary of (Formula presented.), (ii) identifies the interfaces of a partition of a neighborhood of (Formula presented.) solving the minimal partition problem among partitions with same boundary traces. Consequences and sharpness of such results are discussed. The proofs reduce to rather simple and direct arguments based on the exhibition of (global or local) calibrations associated to the minimal network.