The Polya-Tchebotarov problem

TL;DR

This paper characterizes the continuum in the complex plane that minimizes logarithmic capacity containing specific points, with applications to complex analysis and probabilistic estimates.

Contribution

It provides a numerical description of the minimal capacity continuum, improving bounds on related constants and eigenvalues in complex analysis.

Findings

Numerical methods for identifying minimal capacity continua.

Enhanced estimates for the Bloch-Landau constant.

Applications to Brownian motion lifetime and Laplace eigenvalues.

Abstract

We describe the solutions to the problem of identifying the continuum in the complex plane that minimizes the logarithmic capacity among all the continuum that contain a prefixed finite set of points. This description can be implemented numerically and this can be used to improve the estimates on the Bloch-Landau constant and other related problems as the maximal expected lifetime of the Brownian motion on domains of inner radius one or the principal eigenvalue for the Laplace operator on such domains.