We study vector-valued functions that minimise the \(L^\infty \) -norm of their derivatives for prescribed boundary data. We construct a vector-valued, mass minimising 1-current (i.e., a generalised geodesic) in the domain such that all solutions of the problem coincide on its support. Furthermore, this current can be interpreted as a streamline of the solutions. The construction relies on a p-harmonic approximation. In the case of scalar-valued functions, it is closely related to a construction of Evans and Yu (Commun Partial Differ Equ 30:1401–1428, 2005). We therefore obtain an extension of their theory.
