Research on the deformation space of hyperbolic structures on manifolds

流形上双曲结构变形空间研究

• 批准号: 18540080
• 负责人: FUJI Michihiko
• 金额: $ 2.25万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540080/
• 关键词: manifold; cone-structure; hyperbolic structure; deformation; differential equation; discrete group; automaton; Fuchs型微分方程式; 確定特異点; 不確定特異点; 合流操作

The main purpose of this research was to study deformations of a 3-dimensional hyperbolic cone manifold M with non-empty singular set. The head investigator Fujii and the investigator Ochiai constructed an algorithm for solving ordinary differential equations E of Fuchsian type which describe deformations of M. In fact, Fujii and Ochiai succeeded in solving the ordinary differential equations by making use of computers with performing the algorithm. This result was reported in the journal, Publ. Res.Inst Math. Sci 43 (2007). Fujii found some relation between the degeneration of hyperbolic structures on a surface and the confluence of deterministic singular points of ordinary differential equations E concretely. This result was reported in the joumal, Kyushu J.Math. 61 (2007). This result was reported in the workshop, "Actions of hyperbolic groups on specific manifolds and related topics", at Tokyo Metropolitan University, in June 2006. Furthermore, Fujii constructed a geodesic automatic structure for some discrete group G and Computed the growth function of G by making use of a computer program. This result was reported at tie conference, "Riemann surfaces and discrete groups", at Okayama University in January, 2008.The investigator Ue researched the Fukumoto-Furuta invariant for Seifert 3-manifolds and the Neumann-Siebenmann invariant. Ue also studied its spin rational homology cobordism invariance. The investigator Morishita researched the analogy between 3-dimensional topology and Arithmetic. In particular, Morishita studied some relation between SL Chem-Simons theory and Hida-Mazur- theory. The investigator Kawazumi researched Riemann moduli space from the view points of differential geometry. In fact, Kawazumi found some relation between some Chem forms and Johnson homomorphisms for the mapping class groups.

本文主要研究非空奇异集的三维双曲锥流形M的变形问题。首席研究员Fujii和研究员Ochiai构建了一个求解描述m变形的Fuchsian型常微分方程E的算法，实际上，Fujii和Ochiai利用执行该算法的计算机成功地求解了常微分方程。这一结果发表在Publ杂志上。Res.Inst数学。科学43（2007）。Fujii具体地发现了曲面上双曲结构的退化与常微分方程E的确定性奇异点的汇合之间的某种关系。这一结果发表在《九州数学》杂志上。61(2007)。这一结果在2006年6月东京城市大学的“双曲群在特定流形和相关主题上的作用”研讨会上报告。在此基础上，Fujii构造了离散群G的测地线自动结构，并利用计算机程序计算了G的生长函数。这一结果在2008年1月冈山大学召开的“Riemann曲面和离散群”会议上被报道。研究者研究了Seifert 3-流形的Fukumoto-Furuta不变量和Neumann-Siebenmann不变量。我们还研究了它的自旋有理同调协同不变性。研究者Morishita研究了三维拓扑和算术之间的类比。Morishita特别研究了SL Chem-Simons理论和Hida-Mazur-理论之间的关系。研究者Kawazumi从微分几何的角度研究了黎曼模空间。事实上，Kawazumi发现了映射类群的一些化学形式与Johnson同态之间的关系。

项目成果

Growth functions of Fuchsian groups

Fuchsian 群的增长函数

• 发表时间: 2008
• 作者: J.;Itoh;W.;Kiihnel;M.Fujii;K.Kiyoharu;藤井 道彦
• 通讯作者: 藤井 道彦

On singular points of ordinary differential equations on hyperbolic cone-manifolds

双曲锥流形上常微分方程的奇异点

• 发表时间: 2006
• 作者: J.Itoh;W.Kiihonel;藤井 道彦
• 通讯作者: 藤井 道彦

An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions

双曲三锥流形上的调和向量场用超几何函数表示

• 发表时间: 2007
• 期刊: Publ. Res. Inst. Math. Sci. 43
• 作者: Teruaki KITANO;Takayuki MORIFUTI;Teruaki Kitano and Masaaki Suzuki;Takayuki Morifuji;Michihiko Fujii and Hiroyuki Ochiai
• 通讯作者: Michihiko Fujii and Hiroyuki Ochiai

Confluence of singular points of ordinary differential equations of Fuchsian type induced by deformation of two-dimensional hyperbolic cone-manifold structures

二维双曲锥流形结构变形引起的Fuchsian型常微分方程奇异点汇合

• 发表时间: 2007
• 期刊: Kyushu J. Math. 61
• 作者: J. Itoh;K. Kiyohara;M. Fujii
• 通讯作者: M. Fujii