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• W2017757923 abstract "This paper contains some qualitative results about the accessory parameters for punctured spheres with signature. We show that the Fuchsian uniformizing connection, and hence also the accessory parameters, for the surface depends real analytically on moduli. We also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters, Fuchsian groups, Poincare metrics, and covering maps vary continuously under degenerations such as coalescing of punctures. The (branched) holomorphic universal covering group of a Riemann surface S of finite analytic type (g, n) with 2g 2 + n > 0 can be described as the monodromy group of the Schwarzian differential equation where o is a meromorphic connection on S with singularities of very simple type and only at the punctures of S. The connection o depends on 3g 3 + n unknown constants called accessory parameters. Despite numerous attempts (see Hejhal [He2] for some historical remarks), these parameters remain elusive. They can be determined only in the cases where the covering group is elementary or the surface has no moduli (see ?2.2). Nevertheless, the study of the parameters has led to some surprising results. Hejhal [He2] has used variations of accessory parameters to study Poincare series, and in the work of Zograf and Takhtadzhyan [ZT1, ZT2, ZT3] these accessory parameters appear in calculations of the potential for the Weil-Petersson metric on Teichmuller space. In this paper we study the simplest case, that is, n-punctured spheres (g = 0). In this case the connection o is a rational function. We show that o (and hence also the accessory parameters) depends real analytically on moduli and that various objects attached to a uniformization (such as the uniformizing Received by the editors November 30, 1987. 1980MathematicsSubject Classification (1985 Revision). Primary 30F10, 30F35, 32G15, 14H15. Research partially supported by National Science Foundation grants 8401280 and 8120790. () 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page" @default.
• W2017757923 created "2016-06-24" @default.
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• W2017757923 date "1989-02-01" @default.
• W2017757923 modified "2023-09-26" @default.
• W2017757923 title "Accessory parameters for punctured spheres" @default.
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• W2017757923 doi "https://doi.org/10.1090/s0002-9947-1989-0958896-0" @default.
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