Rectifiability of harmonic measure

In the present paper we prove that for any open connected set Ω ⊂ R, n≥ 1 , and any E⊂ ∂Ω with H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω| 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.The second author was supported in part by NSF grant DMS 1361701. The third author has been partially supported by ICMAT Severo Ochoa project SEV-2011- 0087 and he acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089 and NSF UMN MRSEC Seed grant DMR 0212302. The sixth author was supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013) (which also funded the first and fifth authors), by 2014- SGR-75 (Catalonia), MTM2013-44304-P (Spain), and by the Marie Curie ITN MAnET (FP7-607647). The last author was partially supported by the NSF grant DMS-1265549Peer reviewe