Regularity and irregularity of harmonic maps into manifolds

We introduce the definition of minimizing, stationary and weakly harmonic maps. We prove the monotonicity formula and we give some examples of harmonic maps. Then we state the epsilon-regularity theorem for energy minimizing harmonic maps, due to Schoen and Uhlenbeck, and we prove that the singular set of an energy minimizing harmonic map has dimension at most n-3. We also prove the regularity of stationary harmonic maps following the proof of Evans in the case of a target manifold equals to a sphere, and the proof of Bethuel in the case of a general target manifold. Finally we analyze the regularity of weakly harmonic maps. Hélein showed, using the Coulomb gauge frame, that weakly harmonic maps in dimension 2 are still regular maps, but this is not true, in general, in dimension 3 and higher. In fact RIviére, in 1995, proved that there exists a weakly harmonic map from B^3 to S^2 which is singular everywhere in B^3. Our task will be to study in detail the proof proposed by Riviére.