Singularities of intrinsic geometric structures and applications to surfaces and hypersurfaces in Lorentzian spacetimes

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14526
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Yokohama National University
• Principal Investigator: HONDA Atsufumi 横浜国立大学, 大学院工学研究院, 准教授 (90708611)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 曲面 / 超曲面 / 特異点 / 混合型曲面 / 光的点 / 幾何学的不変量 / Bernstein型定理 / 等長変形

Outline of Final Research Achievements

In this study, in order to treat singularities of surfaces and lightlike points on mixed type surfaces in a unified way, degenerate points of metrics and their application to (hyper-)surfaces are investigated. First, as approximations of mixed type surfaces at lightlike points, we obtained a characterization of certain lightlike surfaces in terms of geometric invariants at lightlike points of mixed type surfaces. Moreover, we proved the Bernstein-type theorem for lightlike hypersurfaces, hypersurfaces of zero mean curvature without timelike points, spacelike hypersurfaces of constant mean curvature satisfying a gradient condition. We classified complete null wave fronts whose singular set is non-empty and compact. Furthermore, isometric deformations of surfaces with singularities and their application to curved foldings, a Bour-type isometric deformation theorem for helicoidal surfaces with singularities, and a Fenchel-type theorem for closed curves with singularities are obtained.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

ガウス曲率一定曲面や平均曲率一定曲面は，建築物の構造設計等へ応用されている．そのようなある種の曲率条件を課した曲面には特異点が自然に現れる．その観点から近年，特異点をもつ曲面の理論が注目され，幅広く研究されている．さらに，ローレンツ多様体の平均曲率零曲面は相対論において重要な役割を果たすが，とくに平均曲率零混合型曲面において，第一基本形式がリーマン計量からローレンツ計量に型変化する現象は流体力学としての解釈もある．本研究の結果は純粋に幾何学の理論を追求したものであるが，その建築・物理学への応用も期待される．