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

TL;DR

This paper shows that stable norms on the 2-torus can approximate any strictly convex norm arbitrarily closely, with implications for length spectra in geometric analysis.

Contribution

It proves that the space of stable norms on the 2-torus is dense in the set of strictly convex norms, allowing for controlled approximation with prescribed local behavior.

Findings

Stable norms form a dense subset of strictly convex norms on or 

Constructed sequences of stable norms approximate any given strictly convex norm

Results have implications for length spectrum multiplicities in geometric contexts

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 $\R^2$. In particular, given a strictly convex norm $\Norm_\infty$ on $\R^2$ 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 $\Z^2 \cap \{(a,b) : a^2 + b^2 \leq r \}$ for all $j \geq N$. 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.