Constructing metrics on a $2$-torus with a partially prescribed stable norm

Abstract

A result of Bangert states that the stable norm associated to any Riemannian metric on the $2$-torus $T^2$ is strictly convex. We demonstrate that the space of stable norms associated to metrics on $T^2$ forms a proper dense subset of the space of strictly convex norms on ${\mathbb{R}}^2$. In particular, given a strictly convex norm $\Norm_\infty$ on ${\mathbb{R}}^2$ we construct a sequence $<\Norm_j >_{j=1}^{\infty}$ of stable norms that converge to $\Norm_\infty$ in the topology of compact convergence and have the property that for each $r>0$ there is an $N\equiv N(r)$ such that $\Norm_j$ agrees with $\Norm_\infty$ on ${\mathbb{Z}}^2\cap \{(a,b):a^2+b^2\le r\}$ for all $j\ge N$. Using this result, we are able to derive results on multiplicities which arise in the minimum length spectrum of $2$-tori and in the simple length spectrum of hyperbolic tori.

18 pages