Interpolation by periods in planar domain

Abstract

Let $Ω\subset\mathbb R^2$ be a countably connected domain. To any closed differential form of degree $1$ in $Ω$ with components in $L^2(Ω)$ one associates the sequence of its periods around holes in $Ω$, that is around bounded connected components of $\mathbb R^2\setminus Ω$. For which $Ω$ the collection of such period sequences coincides with $\ell^2$? We give the answer in terms of metric properties of holes in $Ω$.