Uniqueness of selfdual periodic Chern-Simons vortices of topological-type

In analogy with the abelian Maxwell-Higgs model (cf. Jaffe and Taubes in Vortices and monopoles, 1980) we prove that periodic topological-type selfdual vortex-solutions for the Chern-Simons model of Jackiw-Weinberg [Phys Rev Lett 64:2334-2337, 1990] and Hong et al. Phys Rev Lett 64:2230-2233, 1990 are uniquely determined by the location of their vortex points, when the Chern-Simons coupling parameter is sufficiently small. This result follows by a uniqueness and uniform invertibility property established for a related elliptic problem (see Theorem 3.6 and 3.7).