The two-phase problem for harmonic measure in VMO and the chord-arc condition

Let ${\mathrm{\Omega }}^{+}\subset {\mathbb{R}}^{n+1}$ be a bounded $\delta$-Reifenberg flat domain, with $\delta >0$ small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega }}^{-}={\mathbb{R}}^{n+1}\setminus \overline{{\mathrm{\Omega }}^{+}}$ is an NTA domain as well and denote by ${\omega }^{+}$ and ${\omega }^{-}$ the respective harmonic measures of ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$ with poles $p^{\pm }\in {\mathrm{\Omega }}^{\pm }$. In this paper we show that the condition that $\log {\displaystyle \frac{d{\omega }^{-}}{d{\omega }^{+}}}\in VMO({\omega }^{+})$ is equivalent to ${\mathrm{\Omega }}^{+}$ being a chord-arc domain with inner normal belonging to $VMO(H^n{\mathrm{\mid }}_{\mathrm{\partial }{\mathrm{\Omega }}^{+}})$.