Study of the continuations and the spans of an open Riemann surface in view of the thory of functions of several complex variables

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04930
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hiroshima University
• Principal Investigator: SHIBA Masakazu 広島大学, 工学研究科, 名誉教授 (70025469)
• Co-Investigator(Kenkyū-buntansha): 濱野 佐知子 大阪市立大学, 大学院理学研究科, 准教授 (10469588) ; 山口 博史 滋賀大学, 教育学部, 名誉教授 (20025406) ; 増本 誠 山口大学, 大学院創成科学研究科, 教授 (50173761)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: リーマン面 / 等角的埋め込み / 流体力学的微分 / 周期行列 / ジーゲル上半空間 / スパン / 擬凸状領域 / 変分公式

Outline of Final Research Achievements

In our studies preceding the present project we observed the set of all tori into which a given open torus (= an open Riemann surface of genus one) can be conformally embedded and reached the notion of hyperbolic span of the open torus. It can be viewed as a domain functon so that we have a new problem of studying its dependence on the open torus. One of the main results in this project reads: If a family of open bordered tori varying with the parameter in the unit disk forms a pseudoconvex domain in a two-dimensional complex manifold, then the hyperbolic span is a subharmonic function on the unit disk. A condition for the hyperbolic span to be a harmonic function is also given. Some additional results were obtained in this project, which would be useful to generalize the above mentioned theorem to any open Riemann surfaces of finite genus. We studied in particular the period matrices of holomorphic hydrodynamic differentials on such surfaces and obtained a new type of span.

Free Research Field

複素解析学

Academic Significance and Societal Importance of the Research Achievements

開トーラスが1つだけではなく複素径数とともに変化するとき，個々の開トーラスの双曲的スパンが径数領域上の実数値関数として得られるわけだが，その動きを知ることは非常に興味深い（西野 利雄氏の指摘）．私たちの主成果は，少なくとも種数が1の場合においてこの指摘を具体的に定式化し証明したことで，その意義は大きい．
同様の問題を種数が高い開リーマン面についても扱うために，正則な流体力学的微分が定める周期行列を新しい視点に立って定義・考察し古典的周期行列との類似性や相違点を詳しく調べたが，これは従来行われてこなかった研究の方法で意義深い．たとえばその応用のひとつとして新しい型のスパンの概念が得られた．