Faculté des sciences

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... Plus

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    Summary
    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 jN. 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.