  We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the
problem of the finiteness of the induced distance (i.e., the infimum of the
length of horizontal curves that belong to the surface). Recently it has been
proved that for a surface having the topology of a sphere embedded in a tight
co-orientable structure, the distance is always finite. In this paper we study
closed surfaces of genus larger than 1, proving that such surfaces can be
embedded in such a way that the induced distance is finite or infinite. We then
study the structural stability of the finiteness/not-finiteness of the
distance.
