Researches on holomorphic mapings of Riemann surfaces---existence of mappings and conformal invariants

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400140
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Yamaguchi University
• Principal Investigator: MASUMOTO Makoto 山口大学, 大学院創成科学研究科, 教授 (50173761)
• Co-Investigator(Kenkyū-buntansha): 柴 雅和 広島大学, 工学研究科, 名誉教授 (70025469) ; 山田 陽 東京学芸大学, 教育学部, 名誉教授 (60126331) ; 柳原 宏 山口大学, 大学院創成科学研究科, 教授 (30200538) ; 中村 豪 愛知工業大学, 工学部, 教授 (50319208) ; 郷間 知巳 山口大学, 大学院創成科学研究科, 助手 (70253135)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: リーマン面 / 正則写像 / 等角写像 / 極値的長さ / 双曲的長さ / 穴あきトーラス

Outline of Final Research Achievements

A once-holed torus is, by definition, a Riemann surface homeomorphic to a once-punctured torus. The space T of once-holed tori with marked handle is identified with a smooth closed domain in the 3-dimensional euclidean space. Now, fix a Riemann surface Y with marked handle. We investigate the set A of elements X in T which allow holomorphic mappings into Y, and prove that it is a closed domain with Lipschitz boundary and is homeomorphic to T. Moreover, the boundary of A is not smooth in some cases. We also consider the set B of of elements X in T that are
conformally embedded into Y, and obtain similar results for B.

Free Research Field

数物系科学