The distance between homotopy classes of \(S^{1}\)-valued maps in multiply connected domains (Q2480566)

Let \(D\) be a multiply connected domain or embedded compact multiply connected manifold in \({\mathbb R}^n\). The Sobolev space \(H^1(D,S^1)\) can be written as a disjoint union of homotopy classes. The authors consider the following two problems. What is the value of the distance between two different homotopy classes? Is the value achieved? They present a solution for \(D=S^1\). For multiply connected domains in \(\mathbb R^2\), they derive a lower bound for the distance and find conditions that imply the sharpness of the bound. Examples that the lower bound is not always optimal are given.