Mean field equations and domains of first kind

TL;DR

This paper investigates the structure of domains of first and second kind in statistical mechanics, proving openness properties, conditions based on Fourier coefficients, and demonstrating the contractibility of the set of such domains.

Contribution

It introduces new topological and analytical criteria for domains of first kind and establishes their contractibility, advancing understanding in the field.

Findings

Openness property for domains of first kind

Sufficient conditions via Fourier coefficients

Set of first kind domains is contractible

Abstract

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.