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



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.