Development of value distribution theory of Gauss maps of immersed surfaces in space forms and their applications to global property of surfaces

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04840
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kanazawa University
• Principal Investigator: Kawakami Yu 金沢大学, 数物科学系, 准教授 (60532356)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 幾何学 / 複素解析学 / 曲面論 / 値分布論 / 極小曲面 / ガウス写像 / 等角計量 / 解析的延長

Outline of Final Research Achievements

We perform a systematic study of the images of the Gauss maps of complete minimal surfaces in Euclidean 4-space. In particular, we give a geometric interpretation of value-distribution theoretic properties for the Gauss maps of complete minimal surfaces in Euclidean 4-space, for example, the maximal number of exceptional values and unicity theorem. We also provide optimal results of the size of the image under the Gauss map of a complete minimal Lagrangian surface in the complex 2-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean 4-space. We also study the analytic extensions and related global properties of maximal surfaces in the Lorentz-Minkowski 3-space and CMC-1 surfaces in the de-Sitter 3-space.

Free Research Field

微分幾何学，複素解析学，幾何解析

Academic Significance and Societal Importance of the Research Achievements

曲率条件をもつ空間内の曲面は，現実社会の物理的現象としてあらわれるものの数学的モデルになっていることが多い．このような事情から古くから研究され，その成果は物理学や工学など様々な分野の研究に応用されている．本課題の手法は，その実現性をガウス写像という視点で調べるものである．本研究成果は，曲率条件をもつ空間内の曲面の実現性の問題の本質を深めるものであり，幾何学及び複素解析学の研究の発展に意義をもつと考えられる．