A characterisation of $\infty$-harmonic maps in terms of $1$-currents

Abstract

We consider maps between two Riemannian manifolds and study a functional given in terms of the $L^{\infty }$-norm of the derivative. This functional is not differentiable, but we can define critical points with the help of a subdifferential. The resulting notion includes, for example, minimisers in a given homotopy class. We derive a geometric condition equivalent to criticality in this sense. The condition is formulated in terms of a vector-valued $1$-current on the domain manifold, which encapsulates some of the key properties of the critical point. Moreover, this $1$-current is itself a critical point of a generalised mass functional.