  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. It also has maximal
support among all $1$-currents with certain properties.
  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. We therefore obtain an extension of their theory.
