Work

The Teichmüller–Randers metric

Year: 2025

Type: article

Abstract: 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.

Source: Annales de l’institut Fourier

DOI: 10.5802/aif.3716

Authors: Hideki Miyachi, Ken’ichi Ohshika, Athanase Papadopoulos

Institutions: Kanazawa University, Gakushuin University, Centre National de la Recherche Scientifique, Institut de Recherche Mathématique Avancée, Université de Strasbourg

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Institut Henri Poincaré, Japan Society for the Promotion of Science

Awards: Degeneration and collapsing of Kleinian groups; geometry and analysis of the compactification of their defamation spaces, Research on complex analytical structure on Teichmuller space, Geometry of discrete groups and its applications to 3-dimensional topology