Differential geometric research on surfaces admitting singularities and its application

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540100
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo Denki University
• Principal Investigator: KOKUBU Masatoshi 東京電機大学, 工学部, 教授 (50287439)
• Co-Investigator(Renkei-kenkyūsha): UMEHARA Masaaki 東京工業大学, 大学院情報理工学研究科, 教授 (90193945) ; YAMADA Kotaro 東京工業大学, 大学院理工学研究科, 教授 (10221657) ; ROSSMAN Wayne 神戸大学, 大学院理学研究科, 教授 (50284485) ; FUJIMORI Shoichi 岡山大学, 大学院自然科学研究科, 准教授 (00452706) ; YAMAMOTO Ou 東京電機大学, 工学部, 教授 (20291700) ; IRIE Hiroshi 東京電機大学, 未来科学部, 准教授 (30385489)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 微分幾何 / 平均曲率 / ガウス曲率 / 特異点

Outline of Final Research Achievements

We studied surfaces admitting singularities in some kind of three-dimensional manifolds of constant curvature, requiring them to have good properties from the differential-geometric viewpoint. (Note that non-Euclidean space of constant curvature have interesting features beyond our common sense.) Concerning linear Weingarten surfaces in hyperbolic space, we had a global representation formula, criterion for the shape of singularities, and a result on the orientability and the co-orientability. Concerning CMC-1 faces in de Sitter space and maxfaces in Lorentz-Minkowski space, we had results on the orientability and the co-orientability. At the same time, the classification of CMC-1 faces having two ends were obtained, and the classification of maxfaces having three ends were obtained.

Free Research Field

微分幾何学とくに曲面論