Differentiability of the stable norm in codimension one Franz Auer

The real homology of a compact, n-dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm on Hn−1(M,R) is a homogenized version of the Riemannian (n−1)-volume. We study the differentiability properties of the stable norm at points α ∈ Hn−1(M,R). They depend on the position of α with respect to the integer lattice Hn−1(M,Z) in Hn−1(M,R). In particular, we show that the stable norm is differentiable at α if α is totally irrational.