Singularities of surfaces and hypersurfaces in Lorentzian space forms

2022 Fiscal Year Final Research Report

• Project/Area Number: 17H02839
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Yamada Kotaro 東京工業大学, 理学院, 教授 (10221657)
• Co-Investigator(Kenkyū-buntansha): 梅原 雅顕 東京工業大学, 情報理工学院, 教授 (90193945)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 微分幾何学 / ローレンツ多様体 / 直線定理

Outline of Final Research Achievements

The line theorem for a class of surfaces containing zero mean curvature surfaces in Minkowski 3-space, which states that such a surface contains a light-like line if it contains light-like point, is generalized for hypersurfaces in Lorenzian manifolds. As applications, an extension of the Bernstein-type theorem, and a classification of light-like hypersurfaces are obatained.
Under a formulation on analytic extensions of surfaces, the analytic extensions of catenoids in de Sitter 3-space are determined, and proved that they have no further extensions.
Number of isomety class of surfaces having a given space curve as their cuspidal edges, and number of curved paper folding having a given curve as their creases, are determined.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

ローレンツ多様体の因果特性が変化する超曲面はさまざまな例が知られている．また，そのうち解析的延長も持つものも多く知られているが，それらを一般的な視点から記述し，延長不可能性について論じた研究は少なく，学術的に重要なものである．また，これらの対象は自然現象の記述として現れることが多く，他分野への影響も期待される．