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Families of spherical surfaces and harmonic maps

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Abstract

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely \(\mathcal {A}\)-determined map-germs from the plane to the plane can be represented by harmonic maps.

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Acknowledgements

We are very grateful to the referee for valuable suggestions. Part of the work in this paper was carried out while the second author was a visiting professor at Northeastern University, Boston, Massachusetts, USA. He would like to thank Terry Gaffney and David Massey for their hospitality during his visit and FAPESP for financial support with the grant 2016/02701-4. He is partially supported by the grants FAPESP 2014/00304-2 and CNPq 302956/2015-8.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet, Building 303 B, 2800, Kgs. Lyngby, Denmark

    David Brander

  2. Instituto de Ciências Matemáticas e de Computação - USP, Avenida Trabalhador São-Carlense, 400 - Centro, São Carlos, SP, CEP: 13566-590, Brazil

    Farid Tari

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  1. David Brander
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  2. Farid Tari
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Correspondence to David Brander.

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Brander, D., Tari, F. Families of spherical surfaces and harmonic maps. Geom Dedicata 201, 203–225 (2019). https://doi.org/10.1007/s10711-018-0389-3

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  • DOI: https://doi.org/10.1007/s10711-018-0389-3

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