| rationalmapsasschwarzianprimitives | |
Cui Guizhen1 ; Gao Yan2; Rugh Hans Henrik3; Tan Lei4
| |
| 2016 | |
| 发表期刊 | sciencechinamathematics
 |
| ISSN | 1674-7283 |
| 卷号 | 59期号:7页码:1267 |
| 摘要 | We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ 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 φ 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. |
| 语种 | 英语 |
| 文献类型 | 期刊论文 |
| 条目标识符 | http://ir.amss.ac.cn/handle/2S8OKBNM/43475 |
| 专题 | 数学所 |
| 作者单位 | 1.中国科学院数学与系统科学研究院 2.四川大学 3.Faculte des Sciences d'Orsay, Universite Paris-Sud 4.Faculte des Sciences, Laboratoire Angevin de Recherche en Mathematiques, Uniυersite d'Angers |
| 推荐引用方式 GB/T 7714 | Cui Guizhen,Gao Yan,Rugh Hans Henrik,et al. rationalmapsasschwarzianprimitives[J]. sciencechinamathematics,2016,59(7):1267. |
| APA | Cui Guizhen,Gao Yan,Rugh Hans Henrik,&Tan Lei.(2016).rationalmapsasschwarzianprimitives.sciencechinamathematics,59(7),1267. |
| MLA | Cui Guizhen,et al."rationalmapsasschwarzianprimitives".sciencechinamathematics 59.7(2016):1267. |
| 条目包含的文件 | 条目无相关文件。 | |||||
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