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

### Makover, Eran ; Parlier, Hugo ; Sutton, Craig J.

### In: Manuscripta Mathematica, 2012, vol. 139, no. 3-4, p. 515-534

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 R2{/span> . In particular, given a strictly convex norm || · ||∞ on R2{/span> we construct a... More

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- 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 R2{/span> . In particular, given a strictly convex norm || · ||_{∞}on R2{/span> we construct a sequence ⟨∥⋅∥j⟩∞j=1{/span> of stable norms that converge to || · ||_{∞}in the topology of compact convergence and have the property that for each*r*> 0 there is an N≡N(r){/span> such that || · ||_{ j }agrees with || · ||_{∞}on Z2∩{(a,b):a2+b2≤r}{/span> for all*j*≥*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.