The Dirichlet Energy for Maps into the Two Dimensional Sphere

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.