Harmonic Mappings in the Plane

A comprehensive account, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.

Peter Duren (Author)

9780521641210, Cambridge University Press

Hardback, published 29 March 2004

226 pages
22.9 x 15.2 x 1.6 cm, 0.5 kg

'Those who are sensible to the beauty of complex functions and Riemann surfaces will certainly enjoy reading this nicely written … book.' Mathematical Geology

Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

1. Preliminaries
2. Local properties of harmonic mappings
3. Harmonic mappings onto convex regions
4. Harmonic self-mappings of the disk
5. Harmonic univalent functions
6. Extremal problems
7. Mapping problems
8. Additional topics
9. Minimal surfaces
10. Curvature of minimal surfaces
Appendix
References.

Subject Areas: Geometry [PBM], Calculus & mathematical analysis [PBK]