Geometric function theory: a modern view of a classical subject

Abstract

Geometric function theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, nonlinear integrable systems theory and the theory of partial differential equations. This paper surveys, with a view to modern applications, open problems and challenges in this subject. Here we advocate an approach based on the use of the Schottky-Klein prime function within a Schottky model of compact Riemann surfaces.

The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

“…For an overview of the Schottky-Klein prime function and a novel way to compute it, see [6,7]. The P-function (3.17) arises naturally when mapping from an annulus and has been used in the analysis of other Hele-Shaw flow problems with doubly connected domains [34][35][36][37].…”

Secondary Schottky–Klein prime functions associated with multiply connected planar domains

“…The primary S-K prime function on the Schottky double of a planar domain has already been shown in recent years to afford numerous advantages, not to mention key insights, into solving problems in multiply connected domains [1][2][3]. We expect the secondary S-K prime functions explored here to have many similar applications.…”

“…Here, we briefly discuss the numerical computation of the secondary prime functions necessary to generate the examples shown in figures 6-9. First, it is worth reviewing recent progress on the numerical evaluation of the primary S-K prime function (see also references [1][2][3] for additional background). The monograph by Baker [4] cites an infinite product formula only for the evaluation of the (primary) S-K prime function given by…”

Modulo-loxodromic meromorphic functions in $\mathbb C\setminus\{0\}$

“…is called the Schottky-Klein prime function [8]. This function is holomorphic in C * with zero sequence { }, ∈ Z.…”