Harmonic measure distributions of planar domains: a survey

Abstract

Given a domain $\Omega$ in the complex plane and a basepoint $z_0\in \Omega$, the harmonic measure distribution function $h:(0,\infty )\rightarrow [0,1]$ of the pair $(\Omega ,z_0)$ maps each radius $r > 0$ to the harmonic measure of the part of the boundary $\partial \Omega$ within distance r of the basepoint. This function was first introduced by Walden and Ward, inspired by a question posed by Stephenson, as a signature that encodes information about the geometry of $\Omega$. It has subsequently been studied in various works. Two main goals of harmonic measure distribution studies are (1) to understand precisely what can be determined about a domain from its h-function, and (2) given a function $f:(0,\infty )\rightarrow [0,1]$, to determine whether there exists a pair $(\Omega ,z_0)$ that has f as its h-function. In this survey paper, we present key examples of h-functions and summarize results related to these two goals. In particular, we discuss what is known about uniqueness of domains that generate h-functions, necessary conditions and sufficient conditions for a function to be an h-function, asymptotic behavior of h-functions, and convergence results involving h-functions. We also highlight current open problems.