Singular sets for harmonic measure on locally flat domains with locally finite surface measure

TL;DR

This paper demonstrates that in higher dimensions, harmonic measure can be singular with respect to surface measure in certain flat domains, challenging previous assumptions valid in lower dimensions.

Contribution

It constructs examples of Reifenberg flat domains in higher dimensions where harmonic measure is not absolutely continuous with respect to surface measure, showing limitations of classical theorems.

Findings

Harmonic measure can be singular in high-dimensional flat domains.

Classical F. and M. Riesz theorem fails in higher dimensions for these domains.

Existence of domains with finite surface measure but positive harmonic measure on sets of zero surface measure.

Abstract

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $\Omega\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partial\Omega)<\infty$ and a subset $E\subset \partial \Omega$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.