IASM Preprint Series 2022-04: Length minima for an infinite family of filling closed curves on a one-holed torus

ZHONGZI WANG AND YING ZHANG 


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,

on a complete one-holed hyperbolic torus in  its relative Teichmuller space, where a, b are simple closed curves on the one-holed torus which intersect  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 Teichmuller space.