Rectifiability of harmonic measure

TL;DR

This paper proves that for open connected sets in Euclidean space, the absolute continuity of harmonic measure with respect to Hausdorff measure on a subset implies the measure's rectifiability, resolving a longstanding conjecture.

Contribution

It establishes the rectifiability of harmonic measure under absolute continuity conditions, solving a major open problem in the field.

Findings

Harmonic measure's rectifiability follows from absolute continuity on sets with finite Hausdorff measure.

The result confirms the conjecture even in the planar case $n=1$.

Provides a new link between harmonic measure and geometric measure theory.

Abstract

In the present paper we prove that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.