On families of Finsler metrics

TL;DR

This paper explores families of Finsler metrics derived from symmetrisation processes, analyzing their geometric properties and applications to Funk, Hilbert, and Teichmüller geometries, with a focus on geodesics, completeness, and metric shape.

Contribution

It introduces two new families of Finsler metrics from non-symmetric metrics, providing insights into their geometric properties and applications in various geometrical contexts.

Findings

Characterization of geodesics in the new metric families

Conditions for metrics to be Finsler or complete

Descriptions of unit ball shapes in Finsler cases

Abstract

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichm{\"u}ller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichm{\"u}ller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.