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.