This paper is devoted to a study of S-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property (S-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an S-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to S-curves or may be interpreted this way. Existence problem S-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case when both the set of singularities of the external field and the set of fixed points of a class of curves are small (in main case — finite). We also discuss various applications and connections of the theorem.

http://arxiv.org/abs/1112.5713
Complex Variables (math.CV); Mathematical Physics (math-ph)