Accessory parameters in conformal mapping: exploiting isomonodromic tau functions

Abstract

Conformal mappings are important mathematical tools in some applied contexts, e.g. electrostatics and classical fluid dynamics. In order to construct a conformal mapping from a canonical simply connected region to the interior of a circular arc polygon with more than three vertices, the accessory parameter problem arises: In general, the mapping is a solution of a differential equation with unknown parameters which hinder its direct integration. Such parameters can be obtained through approximation techniques with relative small computational effort unless the target domain has an elongated aspect, causing the well known difficulty – the ‘crowding’ phenomenon – to emerge. In this thesis, in the pursuit of calculating the accessory parameters as a Riemann-Hilbert problem, we determine them in terms of isomonodromic tau functions and show how to extract the monodromy information from the geometry of the target domain. We also verify that the tau functions satisfy Toda equations, and this leads to the determination that pre-images of vertex positions are zeros of associated tau functions. We investigate the special case of circular arc quadrilaterals first and in more detail. The isomonodromic tau function then is related to the Painlevé VI transcendent and to certain correlation functions in conformal field theory, yielding asymptotic expansions for the tau function in terms of the monodromy data. We use these expansions to present explicit examples and discuss why the ‘crowding’ phenomenon is not a hindrance for the new framework. In addition, since Schwarz-Christoffel mappings to polygons emerge as a limit when the curvature of the circular arcs goes to zero, we reproduce the well known result for the aspect ratio of rectangles as a function of the accessory parameter. Here, the tau function assumes a closed form in terms of Jacobi theta functions – the Picard solution. Moreover, we use tau function asymptotic expansions to calculate the conformal modules of some trapezoids and find perfect agreement with the literature. We conclude with the investigation of mappings to circular arc polygons with any number of sides, and we comment on the numerical implementation for these cases.