The Teichm{\"u}ller-Randers metric

TL;DR

This paper introduces the Teichmüller-Randers metric, an asymmetric deformation of the classical Teichmüller metric, exploring its properties, geodesic uniqueness, and behavior in specific cases related to measured foliations.

Contribution

It defines a new asymmetric weak metric on Teichmüller space by deforming the classical metric with a differential 1-form and analyzes its fundamental properties.

Findings

Teichmüller-Randers metric is an asymmetric deformation of the Teichmüller metric.

Geodesic uniqueness holds when the 1-form is exact.

The metric is incomplete in any Teichmüller disc when the 1-form relates to extremal length functions.

Abstract

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