Motivated by the asymptotic analysis of double vortex condensates in the Ch
ern-Simons-Higgs theory, we construct a suitable minimizing sequence for a
sharp Sobolev inequality "a la MOSER" for two-dimensional compact manifolds
. As a consequence, we first obtain a direct proof of the sharp character o
f such an inequality. Secondly, and more interestingly, we use such minimiz
ing sequence to show that for the flat torus the corresponding extremal pro
blem attains its infimum.