2015/03/12 by Laurent Hauswirth, Hauswirth, Laurent, Rémy Rodiac +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1503.03648
openalex publication_date 2015/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A \⊂ \ℝ 2 be a smooth doubly connected domain. We\nconsider the Dirichlet energy E(u)=\∫A |\∇ u|2, where u:A\n\→ \ℂ, and look for critical points of this energy with\nprescribed modulus |u|=1 on \∂ A and with prescribed degrees on the\ntwo connected components of \∂ A. This variational problem is a problem\nwith lack of compactness hence we can not use the direct methods of calculus of\nvariations. Our analysis relies on the so-called Hopf differential and on a\nstrong link between this problem and the problem of finding all minimal\nsurfaces bounded by two p covering of circles in parallel planes. We then\nconstruct new immersed minimal surfaces in \ℝ 3 with this property.\nThese surfaces are obtained by bifurcation from a family of p-coverings of\ncatenoids.\n