  We present a renormalization procedure of the Dirichlet Lagrangian for maps
from surfaces with or without boundary into $S^1$ and whose finite energy
critical points are the $S^1-$harmonic maps with isolated singularities. We
give some applications of this renormalization scheme in two different
frameworks. The first application has to do with the renormalization of the
Willmore energy for Lagrangian singular immersions into K\"ahler-Einstein
Surfaces while the second application is dealing with frame energies for
surfaces immersions into Euclidian spaces.
