Stable norms of non-orientable surfaces

Florent Balacheff[1]; Daniel Massart[2]

  • [1] Université de Neuchâtel Institut de mathématiques Rue Émile Argand 11 CP 158 2009 Neuchâtel (Switzerland)
  • [2] Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier UMR 5149 Case Courier 051 Place Eugène Bataillon 34095 Montpellier Cedex 5 (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 4, page 1337-1369
  • ISSN: 0373-0956

Abstract

top
We study the stable norm on the first homology of a closed non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.

How to cite

top

Balacheff, Florent, and Massart, Daniel. "Stable norms of non-orientable surfaces." Annales de l’institut Fourier 58.4 (2008): 1337-1369. <http://eudml.org/doc/10350>.

@article{Balacheff2008,
abstract = {We study the stable norm on the first homology of a closed non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.},
affiliation = {Université de Neuchâtel Institut de mathématiques Rue Émile Argand 11 CP 158 2009 Neuchâtel (Switzerland); Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier UMR 5149 Case Courier 051 Place Eugène Bataillon 34095 Montpellier Cedex 5 (France)},
author = {Balacheff, Florent, Massart, Daniel},
journal = {Annales de l’institut Fourier},
keywords = {Minimizing measures; non-orientable surface; stable norm; minimizing measures},
language = {eng},
number = {4},
pages = {1337-1369},
publisher = {Association des Annales de l’institut Fourier},
title = {Stable norms of non-orientable surfaces},
url = {http://eudml.org/doc/10350},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Balacheff, Florent
AU - Massart, Daniel
TI - Stable norms of non-orientable surfaces
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1337
EP - 1369
AB - We study the stable norm on the first homology of a closed non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.
LA - eng
KW - Minimizing measures; non-orientable surface; stable norm; minimizing measures
UR - http://eudml.org/doc/10350
ER -

References

top
  1. I. Babenko, F. Balacheff, Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006), 347-358 Zbl1082.05509MR2207855
  2. V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1990), 263-286 Zbl0676.53055MR1062758
  3. F. Bonahon, Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) (2001), 1-37 Zbl0996.53029MR1810534
  4. M. J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians, Nonlinearity 8 (1995), 1077-1085 Zbl0845.58023MR1363400
  5. G. Contreras, L. Macarini, Gabriel P. Paternain, Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not. (2004), 361-387 Zbl1086.37032MR2036336
  6. J. Dieudonné, Eléments d’Analyse, 2 (1968), Fasc. XXXI Gauthier-Villars Zbl0189.05502
  7. H. Farkas, I. Kra, Riemann surfaces, 71 (1992), Springer-Verlag, New York Zbl0764.30001MR1139765
  8. A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 267-270 Zbl1052.37514MR1650261
  9. H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351-407 Zbl0289.49044MR348598
  10. M. Gromov, Structures métriques pour les variétés riemanniennes, (1981), 1. CEDIC, Paris Zbl0509.53034MR682063
  11. R. Mañé, Introdução à teoria ergódica, 14 (1983), Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro Zbl0581.28010MR800092
  12. R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity 5 (1992), 623-638 Zbl0799.58030MR1166538
  13. D. Massart, Norme stable des surfaces, Thèse de doctorat. Ecole Normale Supérieure de Lyon (1996) 
  14. D. Massart, Stable norms of surfaces: local structure of the unit ball of rational directions, Geom. Funct. Anal. 7 (1997), 996-1010 Zbl0903.58001MR1487751
  15. John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169-207 Zbl0696.58027MR1109661
  16. G. McShane, I. Rivin, Simple curves on hyperbolic tori, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1523-1528 Zbl0835.53050MR1340065
  17. M. Scharlemann, The complex of curves on nonorientable surfaces, J. London Math. Soc. 25 (1982), 171-184 Zbl0479.57005MR645874
  18. S. Schwartzman, Asymptotic cycles, Ann. of Math. 66 (1957), 270-284 Zbl0207.22603MR88720

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.