Quadratic differentials. (English) Zbl 0547.30001

Strebel, Kurt

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Band 5. Berlin etc.: Springer-Verlag. XII, 184 p. (1984).

This book deals with the theory of quadratic differentials initiated by Teichmüller and developed by the reviewer with subsequent contributions being made by the author and others. It concentrates on providing a highly detailed presentation of the properties of quadratic differentials per se rather than treating the applications to geometric function theory and quasiconformal mappings or the more recently developed connections with foliations.
The first chapter collects some background material on Riemann surfaces. Apart from standard results there is a discussion of free families of homotopy classes of Jordan curves. Chapter two gives the basic definitions for quadratic differentials and chapter three the local trajectory structure at critical points. The fourth chapter deals with the global trajectory structure, primarily for compact surfaces but with some extensions to the general case.
The fifth chapter considers the differential-geometric metric determined by a quadratic differential with particular emphasis on the associated geodesics.
The sixth chapter deals with results arising from the fundamental paper of the reviewer [Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.063)] which related certain general extremal metric problems on a compact Riemann surface to quadratic differentials all of whose structure domains are ring domains and from the equivalent formulation of the author in terms of finding such domains whose modules have prescribed ratios. Related problems and extensions to open surfaces are also presented as well as the rather obvious parallels in the case of reduced modules.
The final chapter deals with some recent considerations of the author and Marden relating to the integral of the imaginary part of the ”square root” of a quadratic differential on a closed path.

Reviewer: J.A.Jenkins

MSC:

30-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable
30-02	Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30F30	Differentials on Riemann surfaces

Keywords:

free families of homotopy classes of Jordan curves; local trajectory structure at critical points; global trajectory structure; geodesics; extremal metric problems; compact Riemann surface; ring domains

Citations:

Zbl 0082.063