Differentiability of the stable norm in codimension one

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 H_n-1(M, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) is a homogenized version of the Riemannian (n-1)-volume. We study the differentiability properties of the stable norm at points α ε H_n-1(M, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]). They depend on the position of α with respect to the integer lattice H_n-1(M, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]) in H_n-1(M, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]). In particular, we show that the stable norm is differentiable at α if α is totally irrational.