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

Let A subset of R-2 be a smooth bounded doubly connected domain. We consider the Dirichlet energy E(u) = integral(A)vertical bar del u(vertical bar)(2), where u : A -> C, and look for critical points of this energy with prescribed modulus vertical bar u vertical bar = 1 on. A and with prescribed degrees on the two connected components of partial derivative 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 quadratic differential and on a strong link between this problem and the problem of finding all minimal surfaces bounded by two p-coverings of circles in parallel planes. We then construct new immersed minimal surfaces in R-3 with this property. These surfaces are obtained by bifurcation from a family of p-coverings of catenoids.