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The Dirichlet Energy for Maps into the Two Dimensional Sphere

  • Mariano Giaquinta
  • Giuseppe Modica
  • Jiří Souček
Chapter
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete / 3. Folge. A Series of Modern Surveys in Mathematics book series (MATHE3, volume 38)

Abstract

In this chapter we begin to develop a natural approach to variational problems for the Dirichlet energy in terms of Cartesian currents. In particular we deal with the regular Dirichlet integral, according to the terminology of Ch. 1, and more specifically, with the Dirichlet integral for mappings from a domain in ℝn or in an oriented n-dimensional Riemannian manifold X into the standard sphere S 2 of ℝ3. In the next chapter we shall discuss the in general non regular Dirichlet energy for mappings from a generic oriented Riemannian manifold X into a generic oriented compact boundaryless Riemannian manifold y.

Keywords

Topological Charge Degree Zero Stereographic Projection Singular Part South Pole 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Mariano Giaquinta
    • 1
  • Giuseppe Modica
    • 2
  • Jiří Souček
    • 3
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità di FirenzeFirenzeItaly
  3. 3.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Čzech Republic

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