Study of value distribution of Gauss maps and its applications to global property of immersed surfaces in space forms

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03463
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kanazawa University
• Principal Investigator: Kawakami Yu 金沢大学, 数物科学系, 准教授 (60532356)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 幾何学 / 解析学 / 曲面論 / 値分布論 / ガウス写像 / 極小曲面 / 平均曲率一定曲面 / 解析的延長

Outline of Final Research Achievements

We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space, As a corollary, we show a Bernstein-type theorem for entire space-like constant mean curvature graphs. Moreover, we show that a notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic completeness. We also a useful extension for that notion of analytic completeness by defining arc-properness of continuous maps. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces or their analytic extensions in the de Sitter 3-space.

Free Research Field

微分幾何学，幾何解析学，複素解析学

Academic Significance and Societal Importance of the Research Achievements

曲率の条件を持つ空間内の曲面は，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となっていることが多い．このことから，曲率の条件を持つ空間内の曲面の性質を調べる研究で得られた成果は，数学にとどまらず，理工学のさまざまな分野の研究に応用されている．本課題の手法は，そのような曲面の性質をガウス写像という視点で調べるものである．研究成果は，曲率を条件に持つ空間内の曲面の実現性の問題の理解を深めるものであり，幾何学及び解析学の研究の発展に意義があると考えられる．