On a sharp Sobolev-type inequality on two-dimensional compact manifolds

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.