Harmonic maps

Harmonic mappings into a closed manifold are defined as critical points of the Dirichlet energy. In this sense, they solve the simplest and most natural variational problem in Riemannian geometry. In the theory, the geometric and analytic aspects are inextricably intertwined, which on one hand extends the avaliable techniques to study harmonic mappings, on the other - is a source of difficulties. Harmonic mappings played an important role in Morrey's solution to the Plateau's problem, methods developed to study their regularity proved very fruitful in the theory of elliptic PDE's.

The lecture will be an elementary introduction to the theory, illustrated with simple examples.