Quadratic differentials and weighted graphs on compact surfaces

We prove that for every simply connected graph Γ embedded in a compact surface R of genus g ≥ 0, whose edges e_kjⁱ carry positive weights w_kjⁱ, there exist a complex structure on R and a Jenkins-Strebel quadratic differential Q(z) dz², whose critical graph Φ_Q complemented, if necessary, by second degree vertices on its edges, is homeomorphic to Γ on R and carries the same set of weights. In other words, every positive simply connected graph on R can be analytically embedded in R. We also discuss a problem on the extremal partition of R relative to such analytical embedding. As a consequence, we establish the existence of systems of disjoint simply connected domains on R with a prescribed combinatorics of their boundaries, which carry proportional harmonic measures on their boundary arcs.