Harmonic maps (Q2865666)

The text (in Russian) consists of lecture notes from a course read by the author in the Scientific-Education Centre of the Steklov Institute in Spring 2008. The main goal is to discuss harmonic maps from Riemann surfaces to Riemannian manifolds through the twistor approach. The text is not sufficiently polished, but is interesting and inspiring as it encompasses several important branches of mathematics from Penrose twistor program and Eells-Wood description of the harmonic maps to Atiyah-Donaldson theorem and Uhlenbeck construction.NEWLINENEWLINENEWLINEThe exposition starts with a general discussion of harmonic maps between Riemann surfaces and their relation to holomorphic maps. Then, the twistor program is exposed. The two almost complex structures on the twistor space of a Riemannian manifold are described, and the method of Rawnsley is explained. Then, the harmonic maps first to the projective spaces and more generally to the Grassmann manifolds are described via holomorphic curves. In the last part of the book harmonic maps to compact Lie groups and their loop spaces are considered. These are related to the gauge theory, construction of instantons, integrable systems and self-dual metrics.