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.