2013/11/27 by Arno B. J. Kuijlaars, Kuijlaars, Arno, Guilherme L. F. Silva +1 · 2 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1311.7026
openalex publication_date 2013/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Curves in the complex plane that satisfy the S-property were first introduced by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s. Rakhmanov recently showed the existence of curves with the S-property in a harmonic external field by means of a max-min variational problem in logarithmic potential theory. This is done in a fairly general setting, which however does not include the important special case of an external field given by the real part of a polynomial of degree greater than or equal to 2. In this paper we give a detailed proof of the existence of a curve with the S-property in the external field given by the real part of a polynomial V, within the collection of all curves that connect two or more pre-assigned directions at infinity in which the real part of V grows. Our method of proof is very much based on the works of Rakhmanov on the max-min variational problem and of Martínez-Finkelshtein and Rakhmanov on critical measures.