The Teichmüller–Randers metric
[La métrique de Teichmüller–Randers]
Miyachi, Hideki1 ; Ohshika, Ken’ichi2 ; Papadopoulos, Athanase3
1 School of Mathematics and Physics College of Science and Engineering Kanazawa University Kakuma-machi, Kanazawa Ishikawa 920-1192 (Japan)
2 Department of Mathematics Gakushuin University Mejiro, Toshima-ku Tokyo 171-8588 (Japan)
3 Institut de Recherche Mathématique Avancée Université de Strasbourg et CNRS 7 rue René Descartes 67084 Strasbourg Cedex (France)
Annales de l'Institut Fourier, Online first, 24 p.

In this paper, we introduce a new asymmetric weak metric on the Teichmüller space of a closed orientable surface with (possibly empty) punctures. This new metric, which we call the Teichmüller–Randers metric, is an asymmetric deformation of the Teichmüller metric and is obtained by adding to the infinitesimal form of the Teichmüller metric a differential 1-form. We study basic properties of the Teichmüller–Randers metric. In the case when the 1-form is exact, any Teichmüller geodesic between two points is also a unique Teichmüller–Randers geodesic between them. A particularly interesting case is when the differential 1-form is the differential of the logarithm of the extremal length function associated with a measured foliation. We show that in this case the Teichmüller–Randers metric is incomplete in any Teichmüller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.

Dans cet article, nous introduisons une nouvelle métrique asymétrique sur l’espace de Teichmüller d’une surface fermée orientable avec ou sans perforations que nous appelons la métrique de Teichmüller–Randers. C’est une déformation asymétrique de la métrique de Teichmüller obtenue en ajoutant une forme différentielle de degré 1 à la forme infinitésimale de cette dernière. Nous étudions les propriétés de base de cette nouvelle métrique. Nous démontrons que dans le cas où la forme différentielle ajoutée est exacte, toute géodésique entre deux points pour la métrique de Teichmüller est aussi une géodésique unique pour la métrique de Teichmüller–Randers. Un cas particulièrement intéressant est celui où la forme différentielle est la différentielle du logarithme de la fonction longueur extrémale associée à un feuilletage mesuré. Nous montrons que dans ce cas la métrique de Teichmüller–Randers restreinte à un disque de Teichmüller quelconque n’est pas complète et nous caractérisons les rayons géodésiques de longueur bornée dans ce disque.

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DOI : 10.5802/aif.3716
Classification : 53B40, 30F60, 32G15
Keywords: Thurston metric, Teichmüller space, Teichmüller disc, Finsler manifold, Randers metric, Teichmüller–Randers metric, extremal length
Mots-clés : métrique de Thurston, espace de Teichmüller, disque de Teichmüller, variété de Finsler, métrique de Randers, métrique de Teichmüller–Randers, longueur extrémale

Miyachi, Hideki 1 ; Ohshika, Ken’ichi 2 ; Papadopoulos, Athanase 3

1 School of Mathematics and Physics College of Science and Engineering Kanazawa University Kakuma-machi, Kanazawa Ishikawa 920-1192 (Japan)
2 Department of Mathematics Gakushuin University Mejiro, Toshima-ku Tokyo 171-8588 (Japan)
3 Institut de Recherche Mathématique Avancée Université de Strasbourg et CNRS 7 rue René Descartes 67084 Strasbourg Cedex (France) 
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Miyachi, Hideki; Ohshika, Ken’ichi; Papadopoulos, Athanase. The Teichmüller–Randers metric. Annales de l'Institut Fourier, Online first, 24 p.

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