The Teichm{ü}llerRanders metric
Abstract
In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}llerRanders metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1form. We study basic properties of the Teichm{ü}llerRanders metric. In the case when the 1form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}llerRanders geodesic between them. A particularly interesting case is when the differential 1form 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{ü}llerRanders metric is incomplete in any Teichm{ü}ller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.16132
 arXiv:
 arXiv:2211.16132
 Bibcode:
 2022arXiv221116132M
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Geometric Topology