Minimizing properties of networks via global and local calibrations

Abstract

In this note we prove that minimal networks enjoy minimizing properties for the length functional. A minimal network is, roughly speaking, a subset of $\mathbb{R}^2$ composed of straight segments joining at triple junctions forming angles equal to $\tfrac23 \pi$; in particular such objects are just critical points of the length functional a priori. We show that a minimal network $\Gamma_*$: i) minimizes mass among currents with coefficients in a suitable group having the same boundary of $\Gamma_*$, ii) identifies the interfaces of a partition of a neighborhood of $\Gamma_*$ 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.