Deformations of the Zolotarev polynomials and Painlevé VI equations (Q2046821)

In this paper, bringing together the connection between the Zolotarev polynomials and the Painlevé VI equations from one hand and the potential theory from other hand, the authors introduce a new type of deformation (the so called iso-harmonic deformation) of annua ardomains with a marked point in the extended complex plane. The introduced iso-harmonic deformation allows the authors to derive a solution of the Painlevé VI equation with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\) in a new way. In particular, starting with the Zolotarev polynomials and varying the elliptic curve that supports the Zolotarev polynomials they obtain a family of elliptic curves with the same property. It turns out that the function \(f(x)\) which determines the position of the zero of the differential of the third kind \(\Omega\) on the pointed family of elliptic curves is a solution of the Painlevé VI equation with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\). Next, the authors involve the obtained result in the potential theory. In particular, they deform an annular domain \(V_h\) (which is the conformal image of the complement of a union of two intervals) and the pole \(c(h)\) of the Green function of \(V_h\) keeping the harmonic measures of \(V_h\) invariant. It turns out that under such a deformation the critical point of the Green function of \(V_h\) with a pole at \(c(h)\) solves the same Painlevé equation VI with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\). The authors call such a deformation an iso-harmonic deformation.