| We use tools from geometric measure theory to catalog fine behavior of harmonic measure on a class of two-sided domains Ω ⊂ Rn in n-dimensional Euclidean space, with n ≥ 3. Assume the interior Ω+ = Ω and exterior Ω− = Rn Ω of Ω are NTA domains, equipped with harmonic measures ω+ and ω−, respectively. We prove that if ω+ and ω − are mutually absolutely continuous and the logarithm of their Radon-Nikodym derivative dω−/dω + has vanishing mean oscillation, then the boundary ∂Ω can be written as a finite disjoint union of sets Γk (1 ≤ k ≤ d) with the following properties. For each Q ∈ Γk, every blow-up of ∂Ω centered at Q is the zero set of a homogeneous harmonic polynomial of degree k which separates space into two connected components; the set Γ1 of “flat points” is relatively open and locally Reifenberg flat with vanishing constant; and the set Γ2∪···∪Γ d of “singularities” has harmonic measure zero. |
