  Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We
consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A
\rightarrow \mathbb{C}$, and look for critical points of this energy with
prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the
two connected components of $\partial A$. This variational problem is a problem
with lack of compactness hence we can not use the direct methods of calculus of
variations. Our analysis relies on the so-called Hopf differential and on a
strong link between this problem and the problem of finding all minimal
surfaces bounded by two $p$ covering of circles in parallel planes. We then
construct new immersed minimal surfaces in $\mathbb{R} ^3$ with this property.
These surfaces are obtained by bifurcation from a family of $p$-coverings of
catenoids.
