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

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

doi:10.1007/s00229-012-0538-1

Back

Journal article

Constructing metrics on a 2-torus 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

13.03.2012

Published in:

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

English 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 || · ||_jagrees 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.

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

Language