Harmonic Mappings in the Plane
A complex valued function of a complex variable is called a harmonic function if it is a solution of the Laplace equation componentwise. Harmonic mappings are then univalent harmonic functions. Any harmonic function can be locally decomposed as a sum of an analytic function and a co-analytic function. If it is a harmonic mapping, then one of those parts is strictly majorized by the other one. If, say, the analytic part prevails, then the mapping is sense preserving. In many aspects, the theory resembles the theory of conformal mappings, but the class of harmonic mappings is much less stable. For example, an inverse of a harmonic mapping typically fails to be harmonic. The theory of harmonic mappings, besides of its own interest, has many applications (e.g., to the theory of minimal surfaces). After several introductory results, the Radó-Kneser-Choquet theorem is presented. This shows that any homeomorphism of the unit circle onto a boundary of a convex domain can be extended to a harmonic mapping of the full disc onto the closure of the domain. Of course, the extension is nothing more than the solution of the Dirichlet problem but the main point of the theorem is to show that the solution is univalent. The shear construction, which leads to interesting examples of explicit harmonic mappings, is described and applied. Another class of explicit harmonic mappings with dilatation of the type of a Blaschke product is used to map the disc onto a convex polygon. The harmonic Koebe function is a very interesting mapping, which is extremal for many problems.
A part of the book is devoted to a study of analogues of the Riemann mapping theorem. The situation is much more complicated here than in the conformal case. It is also interesting to observe what is known for multiply connected domains. Estimates in Hardy spaces are represented by a few results. Many nice results relating coefficients of the Taylor expansion of the analytic and coanalytic part with the image of the mapping are formulated. The last part of the book shows how the theory can be applied to minimal surface problems. The Weierstrass-Enneper representation of minimal surfaces is explained and minimal graphs are studied. The theory of harmonic mappings is applied to curvature estimates of minimal surfaces. The roots of the theory of harmonic mappings can be considered as classical, its development is fluent, and interesting problems still wait for their solution. The book by Peter Duren is the first comprehensive treatment of the topic. Any friend of complex analysis will admire the beauty of this extension of the theory so nicely presented in the volume.