Minima of geodeics length functions for non-uniform filling

Kerckhoff proved that the geodesic length function ${\mathrm{\ell }}_{\mathrm{\Omega }}$ of a filling $\mathrm{\Omega }$ on $S_g$ attains a unique minimum in Teichmüller space. Recent work of Ernesto Girondo et al. computed these minima for uniform fillings using the algebraic machinery of dessins d’enfants and Grothendieck-Belyi surfaces. We present an elementary optimization approach for $4$-regular topological uniform fillings, bypassing this framework. Furthermore, we analyze two special classes of non-uniform $4$-regular fillings using fat graphs and optimization techniques. We explicitly compute their minima and prove that in both classes, the minimum of these length functions is attained at a triangle surface.