Explicit finite geodesic set for distances on genus ≥ 2 surfaces

Determine, for any compact hyperbolic Riemann surface of genus g ≥ 2 and any pair of points on the surface, the explicit finite set of geodesic paths connecting the points whose length minimum equals the intrinsic distance between them as defined by the hyperbolic metric. The goal is to specify this finite set concretely so that the distance formula involving the infimum over the Fuchsian group actions can be evaluated in finite time for arbitrary genus g ≥ 2 surfaces.

Background

Distances on compact hyperbolic Riemann surfaces of genus g ≥ 2 are defined via an infimum over infinitely many geodesics induced by the Fuchsian group action on the hyperbolic plane. Although it is known (by Proposition 2.1 in Lu and Meng, 2020) that the infimum is actually attained as a minimum over a finite set of geodesics, the explicit description of this finite set is unknown in general. This paper provides a computable formula and an algorithm for a broad class of quotient surfaces obtained from convex fundamental polygons, but the general problem of identifying the finite set for arbitrary genus g ≥ 2 surfaces remains unresolved.

The explicit identification of this finite set is crucial for practical computation of distances in many applications across mathematics and physics where Riemann surfaces of higher genus play a central role.