Hedlund metrics and the stable norm

Abstract

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H 1 ( 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 H 1 ( 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 H 1 ( M , R ) , whose vertices are rational with respect to the lattice of integer classes in H 1 ( 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.