Gribov copies of the vacuum in two dimensional compact U (1) lattice g
auge models are constructed. On the basis of this a gauge fixing algor
ithm is developped, wich finds the minimum of the sum of link field sq
uares. Numerical experience in a two dimensional Higgs model with fixe
d length scalar field is reported and the extension to three and four
dimensional U(1) and four dimensional SU(2) gauge theories is briefly
discussed.