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

Motivated by the asymptotic analysis of double vortex condensates in the Chern-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 of such an inequality. Secondly, and more interestingly, we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum.