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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三维双曲锥流形M的变形。首席研究员Fujii和研究员Ochiai构建了一个求解Fuchsian型常微分方程组的算法，该算法描述了M的变形。该算法发表在《跨学科信息科学》杂志9(2003)上。Fujii发现了双曲纽结上双曲结构的退化与椭圆曲线上的一些有理点之间的某种联系。这一结果发表在《数学杂志》上。京都大学，45(2005)。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。这一结果发表在2004年12月在东京工业大学举行的“黎曼表面和不连续群”会议上。此外，Fujii发现这些有理点可能位于某个圆上，该圆构成了模曲线的某个基本区域的边界。这一结果于2005年11月在大阪产业大学举行的“拓扑学与计算机2005”会议上发表。UE证明了Seifert三维流形上的Fukumoto-Furuta不变量与Neumann-Siebenmann不变量重合，并证明了它的自旋有理同调协边不变性。得到了Seifert三维流形通过对三维球面上的纽结进行Dehn运算而得到的条件。调查者Saito成功地给出了非阿基米德局部场上的可容许表示的交织算子的具体描述。