On weakly harmonic maps and Noether harmonic maps from a Riemann surface into a Riemannian manifold

Abstract

I want to present some results on the regularity for harmonic maps between a surface of dimension two and a Riemannian manifold. First of all, we recall what harmonic maps between Riemannian manifolds are. Let (M, g) and (N , g) be two Riemannian manifolds of dimension m and n respectively, and assume that N is compact and is isometrically embedded in some Euclidean space R (which is always possible thanks to the Nash–Moser theorem). We introduce the Dirichlet functional on the set of maps between M and N . To do this we define the energy density of a map u from M into N at any point x of M by e(u)(x) = 1 2hij [u(x)]g (x)uαu j β