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

TL;DR

This paper studies harmonic maps with fixed boundary degrees on an annulus, linking the problem to minimal surfaces, and constructs new immersed minimal surfaces via bifurcation from catenoids.

Contribution

It introduces a novel connection between harmonic maps with boundary conditions and minimal surface theory, leading to the construction of new immersed minimal surfaces.

Findings

Identification of critical points of Dirichlet energy with prescribed boundary degrees

Establishment of a link between harmonic maps and minimal surfaces bounded by circle coverings

Construction of new immersed minimal surfaces through bifurcation analysis

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