Interpolation by periods in planar domain

TL;DR

This paper investigates the relationship between differential forms in planar domains and their period sequences, characterizing when these sequences form an $ ext{ exttt{l}}^2$ space based on the geometric properties of the domain's holes.

Contribution

It provides a characterization of domains in the plane for which the period sequences of differential forms with $L^2$ components are exactly the $ ext{ exttt{l}}^2$ sequences, linking geometric properties to functional analysis.

Findings

Identifies metric conditions on holes for period sequences to be in $ ext{ exttt{l}}^2$

Connects geometric properties of domain holes with functional analytic properties of differential forms

Provides criteria for when period sequences form an $ ext{ exttt{l}}^2$ space

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 $\Omega$.