Mathematical Research Letters

Volume 4 (1997)

Number 6

On the structure of the stable norm of periodic metrics

Pages: 791 – 808



D. Burago

S. Ivanov

B. Kleiner


We study the differentiability of the stable norm $\norm$ associated with a ${\Bbb Z}^n$ periodic metric on ${\Bbb R}^n$. Extending one of the main results of \cite{Ba2}, we prove that if $p\in {\Bbb R}^n$ and the coordinates of $p$ are linearly independent over $\Bbb Q$, then there is a linear 2-plane $V$ containing $p$ such that the restriction of $\norm$ to $V$ is differentiable at $p$. We construct examples where $\norm$ it is not differentiable at a point with coordinates linearly independent over $\Bbb Q$.

Full Text (PDF format)