Length minima for an infinite family of filling closed curves on a one-holed torus

TL;DR

This paper explicitly determines the minimal lengths and points for a family of filling closed curves on a one-holed hyperbolic torus, providing concrete examples for the problem of minimizing geodesic lengths in Teichmüller space.

Contribution

It explicitly finds the minima and minimum points of geodesic length functions for a specific family of filling curves on a one-holed hyperbolic torus, advancing understanding of length minimization in Teichmüller space.

Findings

Explicit minima and minimum points for the length functions.

Concrete examples for length minimization problems.

Enhanced understanding of geodesic length behavior on hyperbolic surfaces.

Abstract

We explicitly find the minima as well as the minimum points of the geodesic length functions for the family of filling (hence non-simple) closed curves, $a^2b^n$ ($n\ge 3$), on a complete one-holed hyperbolic torus in its relative Teichm\"uller space, where $a, b$ are simple closed curves on the one-holed torus which intersect exactly once transversely. This provides concrete examples for the problem to minimize the geodesic length of a fixed filling closed curve on a complete hyperbolic surface of finite type in its relative Teichm\"uller space.