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We explicitly find the minima as well as the minimum points of the geodesiclength 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 relativeTeichm\"uller space, where $a, b$ are simple closed curves on the one-holedtorus which intersect exactly once transversely. This provides concreteexamples for the problem to minimize the geodesic length of a fixed fillingclosed curve on a complete hyperbolic surface of finite type in its relativeTeichm\"uller space.

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