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

Makover, Eran
Department of Mathematics, Central Connecticut State University, New Britain, USA

Parlier, Hugo
Department of Mathematics, University of Fribourg, Switzerland

Sutton, Craig J.
Department of Mathematics, Dartmouth College, Hanover, USA
Published in:
 Manuscripta Mathematica.  2012, vol. 139, no. 34, p. 515534
English
A result of Bangert states that the stable norm associated to any Riemannian metric on the 2torus 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 2tori and in the simple length spectrum of hyperbolic tori.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/302721
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