On computation of capacities and conformal invariants

TL;DR

This paper surveys methods for computing conformal capacities and invariants of planar domains, emphasizing recent boundary integral techniques and their applications to various conformal invariants.

Contribution

It introduces a unified boundary integral approach for computing multiple conformal invariants, highlighting recent developments and applications in potential theory.

Findings

Effective boundary integral method for conformal capacity computation

Application of boundary integral equations to harmonic measure and hyperbolic capacity

Compilation of extensive bibliography on constructive complex analysis

Abstract

We give a survey of computation of the conformal capacity of planar condensers, generalized capacity, and logarithmic capacity with emphasis on our recent work 2020-2025. We also discuss some applications of our method based on the boundary integral equation with the generalized Neumann kernel to the computation of several other conformal invariants: harmonic measure, modulus of a quadrilateral, reduced modulus, hyperbolic capacity, and elliptic capacity. Here the solution of mixed Dirichlet-Neumann boundary value problem for the Laplace equation has a key role. At the end of the paper we give a topicwise structured list to our extensive bibliography on constructive complex analysis and potential theory.