Hedlund metrics and the stable norm

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R)H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H1(M,R)H1(M,R) are stable norms of a Riemannian metric on M. If the dimension of M is at least three, I. Babenko and F. Balacheff proved in [I. Babenko, F. Balacheff, Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119 (3) (2006) 347–358] that every polyhedral norm ball in H1(M,R)H1(M,R), whose vertices are rational with respect to the lattice of integer classes in H1(M,R)H1(M,R), is the stable norm ball of a Riemannian metric on M. This metric can even be chosen to be conformally equivalent to any given metric. In [I. Babenko, F. Balacheff, Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119 (3) (2006) 347–358], the stable norm induced by the constructed metric is computed by comparing the metric with a polyhedral one. Here we present an alternative construction for the metric, which remains in the geometric framework of smooth Riemannian metrics.