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Families of spherical surfaces and harmonic maps

  • David Brander
  • Farid Tari
Original paper
  • 42 Downloads

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.

Keywords

Bifurcations Differential geometry Discriminants Integrable systems Loop groups Parallels Spherical surfaces Constant Gauss curvature Singularities Cauchy problem Wave fronts 

Mathematics Subject Classification (2010)

Primary 53A05 53C43 Secondary 53C42 57R45 

Notes

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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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark
  2. 2.Instituto de Ciências Matemáticas e de Computação - USPSão CarlosBrazil

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