Rational maps as Schwarzian primitives

doi:10.1007/s11425-016-5140-7

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	Rational maps as Schwarzian primitives
	Cui GuiZhen1 ; Gao Yan 2; Rugh, Hans Henrik 3; Tan Lei 4
	2016-07-01
发表期刊	SCIENCE CHINA-MATHEMATICS
ISSN	1674-7283
卷号	59 期号:7 页码:1267-1284
摘要	We examine when a meromorphic quadratic differential I center dot with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of I center dot around each pole c, the most singular term should take the form (1 - d (2))/(2(z - c)(2)), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles (i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko (2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by I center dot outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative. Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
关键词	Schwarzian derivatives rational maps critical points meromorphic quadratic differentials
DOI	10.1007/s11425-016-5140-7
语种	英语
资助项目	National Natural Science Foundation of China[11125106] ; National Natural Science Foundation of China[11501383] ; Project LAMBDA[ANR-13-BS01-0002]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000379083700003
出版者	SCIENCE PRESS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/23044
专题	数学所
通讯作者	Gao Yan
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 2.Sichuan Univ, Dept Math, Chengdu 610065, Peoples R China 3.Univ Paris 11, Fac Sci Orsay, F-91405 Paris, France 4.Univ Angers, Fac Sci, Lab Angevin Rech Math, F-49045 Angers, France
推荐引用方式 GB/T 7714	Cui GuiZhen,Gao Yan,Rugh, Hans Henrik,et al. Rational maps as Schwarzian primitives[J]. SCIENCE CHINA-MATHEMATICS,2016,59(7):1267-1284.
APA	Cui GuiZhen,Gao Yan,Rugh, Hans Henrik,&Tan Lei.(2016).Rational maps as Schwarzian primitives.SCIENCE CHINA-MATHEMATICS,59(7),1267-1284.
MLA	Cui GuiZhen,et al."Rational maps as Schwarzian primitives".SCIENCE CHINA-MATHEMATICS 59.7(2016):1267-1284.