Mutual absolute continuity of interior and exterior harmonic measure   implies rectifiability
  We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. 
Jonas Azzam, Mihalis Mourgoglou, and Xavier Tolsa
math.CA math.AP math.MG