Fuchsian differential equations. With special emphasis on Gauss-Schwarz theory.

*(English)*Zbl 0618.35001
A Publication of the Max-Planck-Institut für Mathematik, Bonn. Adviser: Friedrich Hirzebruch. Aspects of Mathematics. Aspekte der Mathematik, Vol. E11. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. XIV, 215 p.; DM 48.00 (1987).

The author studies linear ordinary differential equations and systems of linear partial differential equations with finite dimensional solution spaces in the complex analytic category, concentrating exclusively on the Gauss-Schwarz theory of the hypergeometric differential equation. This theory is presented as part of the theory of orbifolds, i.e. to each orbifold uniformized by a symmetric space, one attempts to associate a uniformizing differential equation. In order to construct orbifolds, recent results of R. Kobayashi are used in a crucial way. Thus new Fuchsian differential equations in two variables with remarkable properties are constructed.

Part I covers hypergeometric differential equations, the Riemann and Riemann-Hilbert problems, Schwarzian derivatives and the Gauss-Schwarz theory for equations in one variables in chapters 1 to 5 respectively. Chapters 6 to 10 of Part II does the same for equations in several variables. Ch. 11 deals with reflection groups and ch. 12 presents new differential equations.

Part I covers hypergeometric differential equations, the Riemann and Riemann-Hilbert problems, Schwarzian derivatives and the Gauss-Schwarz theory for equations in one variables in chapters 1 to 5 respectively. Chapters 6 to 10 of Part II does the same for equations in several variables. Ch. 11 deals with reflection groups and ch. 12 presents new differential equations.

Reviewer: R.Vaillancourt

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

35A20 | Analyticity in context of PDEs |

35Q15 | Riemann-Hilbert problems in context of PDEs |