Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids

Let $A\subset {\mathbb{R}}^2$ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)={\int }_A\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2$, where $u:A \rightarrow $\mathbb{C}$$, and look for critical points of this energy with prescribed modulus $\mathrm{\mid }u\mathrm{\mid }=1$ on $\mathrm{\partial }A$ and with prescribed degrees on the two connected components of $\mathrm{\partial }A$. 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.