Hyperbolic structures on manifolds and their deformations

流形上的双曲结构及其变形

基本信息

批准号：
15540069
负责人：
FUJII Michihiko
金额：
$ 2.11万
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：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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 of Fuchsian type which describe deformations of M. This algorithm was reported in the journal, Interdisciplinary Information Sciences 9(2003). Fujii found some relation between the degeneration of hyperbolic structures on a hyperbolic knot and some rational points of an elliptic curve. This result was reported in the journal, J.Math.Kyoto Univ.45(2005). Fujii succeeded in drawing such a rational point on the modular curve of the elliptic curve. This result was reported in the conference, "Riemann Surfaces and Discontinuous Groups" at Tokyo Institute of Technology in December 2004. Furthermore, Fujii found the possibility that these rational points are located on some circle which constitute the boundary of some fundamental region of the modular curve. This result was reported at the conference, "Topology and Computer 2005", at Osaka Sangyo University in November 2005.The investigator Ue proved that the Fukumoto-Furuta invariant for Seifert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. Ue obtained the conditions for Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The investigator Saito succeeded to give a concrete description of admissible representations on non-Archimedes local fields in terms of intertwining operators.

本研究的主要目的是研究具有非空奇异集的3维双曲锥流形M的变形。首席研究员藤井和研究员落合构建了一种求解描述 M 变形的 Fuchsian 型常微分方程的算法。该算法发表在《跨学科信息科学》杂志 9（2003）上。藤井发现双曲结上双曲结构的退化与椭圆曲线的一些有理点之间存在某种关系。该结果发表在J.Math.Kyoto Univ.45(2005)杂志上。藤井成功地在椭圆曲线的模曲线上画出了这样一个有理点。这一结果在 2004 年 12 月于东京工业大学举行的“黎曼曲面和不连续群”会议上发表。此外，藤井还发现这些有理点可能位于某个圆上，而该圆构成了模的某些基本区域的边界。曲线。这一结果于 2005 年 11 月在大阪产业大学举行的“Topology and Computer 2005”会议上发表。 Ue 研究员证明了 Seifert 3 流形的 Fukumoto-Furuta 不变量与 Neumann-Siebenmann 不变量一致，并证明了其自旋有理同源配边不变性。 Ue 获得了通过对 3 球体中的结进行 Dehn 手术获得 Seifert 3 流形的条件。研究者 Saito 成功地根据交织算子对非阿基米德局部域上可接受的表示进行了具体描述。