  In the present work we study the behavior of sequences of smooth global
isothermic immersions of a given closed surface and having a uniformly bounded
total curvature. We prove that, if the conformal class of this sequence is
bounded in the Moduli space of the surface, it weakly converges in W^{2,2} away
from finitely many points, modulo extraction of a subsequence, to a possibly
branched weak isothermic immersion of this surface. Moreover, if this limit
happens to be smooth away from the branched points, we give an optimal
description of the possible loss of strong compactness of such a subsequence by
proving that, beside possibly finitely many atomic concentrations, the defect
measure associated to the L^2 norm of the second fundamental form is
"transported" along exceptional directions given by some holomorphic quadratic
forms associated the limiting surface. We give examples where these losses of
compactness, invariant along such exceptional directions, eventually happen.
