From 2D Toda hierarchy to conformal map for domains of Riemann sphere

TL;DR

This paper explores the connection between the dispersionless 2D Toda hierarchy and conformal maps of Riemann sphere domains, providing conditions for the convergence of series used in calculating these maps.

Contribution

It introduces a sufficient condition for the convergence of Taylor series in the context of conformal mapping via integrable systems, advancing the computational methods for univalent maps.

Findings

Derived a convergence criterion for the Taylor series of the string solution.

Provided a method to compute conformal maps from the unit disk to arbitrary domains.

Linked integrable systems with classical conformal mapping problems.

Abstract

In recent works [hep-th/9909147, hep-th/0005259] was found a wonderful correlation between integrable systems and meromorphic functions. They reduce a problem of effictivisation of Riemann theorem about conformal maps to calculation of a string solution of dispersionless limit of the 2D Toda hierarchy. In [math.CV/0103136] was found a recurrent formulas for coeffciens of Taylor series of the string solution. This gives, in particular, a method for calculation of the univalent conformal map from the until disk to an arbitrary domain, described by its harmonic moments. In the present paper we investigate some properties of these formulas. In particular, we find a sufficient condition for convergence of the Taylor series for the string solution of dispersionless limit of 2D Toda hierarchy.