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Mean field equations and domains of first kind

In this paper we are interested in understanding the structure of domains of first and second kind, a concept motivated by problems in statistical mechanics. We prove some openness property for domains of first kind with respect to a suitable topology, as well as some sufficient condition for a simply connected domain to be of first kind in terms of the Fourier coefficients of the Riemann map. Finally, we show that the set of simply connected domains of first kind is contractible.