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

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Abstract

Let \(A \subset \mathbb {R} ^2 \) be a smooth bounded 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 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 \(\mathbb {R}^3\) with this property. These surfaces are obtained by bifurcation from a family of p-coverings of catenoids.

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Acknowledgments

The second author wishes to warmly thank Etienne Sandier for his help and for constant support during the elaboration of this paper. He also wants to thank Mickaël Dos Santos, Yuxin Ge, Laurent Mazet, Rabah Souam, Eric Toubiana and his colleague Peng Zhang for very useful discussions on this paper.

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Authors and Affiliations

  1. Université de Marne-la-Vallée, cité Descartes-5 boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Laurent Hauswirth

  2. Université Paris-Est Créteil, 61 avenue du général de Gaulle, 94010, Créteil Cedex, France

    Rémy Rodiac

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  1. Laurent Hauswirth
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  2. Rémy Rodiac
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Correspondence to Rémy Rodiac.

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Communicated by A. Malchiodi.

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Hauswirth, L., Rodiac, R. Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids. Calc. Var. 55, 120 (2016). https://doi.org/10.1007/s00526-016-1059-7

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