Two-dimensional Geometric Variational Problems

This monograph addresses variational problems for mappings from a surface equipped with a conformal structure into Euclidean space or a Riemannian manifold. It is assumed that the variational problems are invariant under conformal reparametrizations of the domain. Solutions to such variational problems consist of conformal mappings between surfaces, minimal surfaces in Riemannian manifolds, harmonic maps from a surface into a Riemannian manifold, and solutions of prescribed mean curvature equations. A general theory of such variational problems is presented, proving existence and regularity theorems with particular conceptual emphasis on the geometric aspects of the theory and thoroughly investigating the connections with complex analysis. The approach is purely parametric, and consequently, does not address the question of geometric regularity of the solutions (immersion and embeddedness properties).

Table of Contents:
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.