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

Abstract

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 $u:A\to \mathbb{C}$, 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.

48 pages, 2 figures

References in corpus (1)

• On compound vortices in a two-component Ginzburg-Landau functional