Differentiability of the stable norm in codimension one
Abstract
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,R) is a homogenized version of the Riemannian (n-1)-volume. We study the differentiability properties of the stable norm at points alpha in H_{n-1}(M,R). They depend on the position of alpha with respect to the integer lattice H_{n-1}(M,Z) in H_{n-1}(M,R). In particular, we show that the stable norm is differentiable at alpha if alpha is totally irrational.
Cite
@article{arxiv.math/0403235,
title = {Differentiability of the stable norm in codimension one},
author = {Franz Auer and Victor Bangert},
journal= {arXiv preprint arXiv:math/0403235},
year = {2007}
}
Comments
28 pages LaTeX. Submitted to American Journal of Mathematics; Reference added. Minor inaccuracy corrected; Revised version for publication after referee report. Proof of Propositon 2.4 added. Some inaccuracies corrected and some unclear formulations changed