Deformations of the Zolotarev polynomials and Painlevé VI equations

Abstract

The aim of this paper is to introduce new type of deformations of domains in the extended complex plane with a marked point and associated Green functions, the so-called iso-harmonic deformations in the first nontrivial case of doubly connected domains and to study their isomonodromic properties. We start with the Zolotarev polynomials, which are a particular case of generalized Chebyshev polynomials, namely minimal polynomials on two intervals. We introduce a deformation of elliptic curves which support Zolotarev polynomials and relate it to the Painlevé VI equations. Then, we transport these considerations into the realm of potential theory of annular domains. We deform these domains and the poles of the associated Green functions in a specific new way, by keeping invariant the corresponding harmonic measure of the boundary circles. We deduce that the critical points of the Green functions under such deformations solve a Painlevé VI equation.