Mathematical Research Letters

Volume 4, Issue 6, November 1997 pp. 791-808.

On the structure of the stable norm of periodic metrics

Authors: D. Burago, S. Ivanov, and B. Kleiner
Author institution: Pennsylvania State University, and Steklov Institute for Mathematics

Summary: 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 $\Bbb Q$.

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