Two-dimensional Geometric Variational Problems

Part 1 Examples, definitions, and elementary results: Plateau's problem; two dimensional conformally invariant variational problems; harmonic maps, conformal maps and holomorphic quadratic differentials; some applications of holomorphic quadratic differentials; surfaces in R3; the Gauss map. Part 2 Regularity and uniqueness results: harmonic co-ordinates; uniqueness of harmonic maps; continuity of weak solutions; removability of isolated singularities; higher regularity; the Hartmann-Wintner lemma and some of its consequences; asymptotic expansions at branch points; estimates from below for the functional determinant of univarlent harmonic mappings. Part 3 Conformal representation: conformal representation of surfaces homeomorphic to S2 and circular domains and closed surfaces of higher genus. Part 4 Existence results: the local existence theorem - an easy proof of the existence of energy minimizing maps; the general existence theorem; corollaries and consequences of the general existence theorem - boundary conditions; nonexistence results - existence of maps with holomorphic quadratic differentials; another proof of the existence of unstable minimal surfaces; the Plateau-Douglas problem in Riemannian manifolds. Part 5 Harmonic maps between surfaces: the existence of harmonic diffeomorphisms; local computations; consequences for nonpositively curved image metrics; harmonic diffeomorphisms; Kneser's Theorem; harmonic maps and branched coverings. Part 6 Harmonic maps and Teichmuller spaces: the basic definitions; the topological and differentiable structure of Teichmuller's Theorem; the complex structure; the energy as a function of the domain metric; the metric structure; the Weil-Petersson metric; Kahler property.