In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a convex function of the twist parameter about that geodesic (Proposition 3.3.11).
Does a similar result hold in the case of surfaces with cusps?
In this case, Buser's argumentation breaks down as many of the lengths of the curves he examines become infinite and his geometric picture is no longer useful. In fact, As there are no geodesics inside of a cusp, you cannot even define the transversal for surfaces with cusps in the same way as for surfaces without. Any suggestions would be helpful.