Minimal rays on surfaces of genus greater than one -- Part II

TL;DR

This paper investigates the behavior of minimal rays on surfaces of genus greater than one with Finsler metrics, showing that rays sharing a non-fixed asymptotic direction intersect minimally, influencing the structure of geodesics and Busemann functions.

Contribution

It proves that minimal rays with a common asymptotic direction, not fixed by deck transformations, intersect only at their start, revealing new geometric properties of such surfaces.

Findings

Minimal rays with shared asymptotic directions intersect only at initial points.

The structure of minimal geodesics is constrained by asymptotic directions.

Busemann functions exhibit specific properties related to these minimal rays.

Abstract

We consider any Finsler metric on a closed, orientable surface of genus greater than one. H. M. Morse proved that we can associate an asymptotic direction to minimal rays in the universal cover (in the Poincar\'e disc: a point on the unit circle). We prove here that, if two minimal rays have a common asymptotic direction, which is not a fixed point of the group of deck transformations, then the two rays can intersect at most in a common initial point. This has strong consequences for the structure of the set of minimal geodesics, as well as for the set of Busemann functions associated to the Finsler metric.