Abstract

For C¹ maps between Riemannian manifolds, we can define their Dirichlet energy. When the target manifold is R^m with the Euclidean metric, critical points of the Dirichlet energy are harmonic, and so their regularity is easy to establish. However in general this is not the case as the curvature causes the Euler-Lagrange equations to become non-linear. In this case, the regularity theory is more subtle, and the possibility of a small singular set arises. In this talk I will discuss both cases, working up to the statement of the Schoen-Uhlenbeck theorem, which is the main result for establishing the regularity theory of such maps. We will then discuss the size of the singular set which can arise.