Interpolation by periods in planar domain

Abstract

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

87 pages, 13 figures