  There is a natural link between (multi-)curves that fill up a closed oriented
surface and dessins d'enfants. We use this approach to exhibit explicitly the
minima of the geodesic length function of a kind of curves (uniform filling
curves) which include those that admit a homotopy equivalent representative
such that all self-intersection points as well as all faces of their complement
have the same multiplicity. We show that these minima are attained at the
Grothendieck-Belyi surfaces determined by a natural dessin d'enfant associated
to these filling curves. In particular they are all Riemann surfaces defined
over number fields.
