Principal distributions on surfaces in various spaces

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05221
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kumamoto University
• Principal Investigator: Ando Naoya 熊本大学, 大学院先端科学研究部(理), 准教授 (50359965)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 零平均曲率ベクトル / 等方性 / 正則4次微分 / ツイスター・リフト / 共形Gauss写像 / Willmore曲面 / Gauss写像

Outline of Final Research Achievements

I studied isotropicity of space-like or time-like surfaces with zero mean curvature vector in neutral or Lorentzian 4-manifolds. In particular, in the neutral case, the isotropicity of time-like surfaces with zero mean curvature vector does not necessarily mean horizontality of the twistor lifts and the covariant derivatives of the lifts of the conformal Gauss maps of time-like minimal surfaces in the 3-dimensional flat Lorentzian space form are light-like. In the Lorentzian case, I defined isotropicity and obtained related results.
I obtained analogues of holomorphicity of the Gauss maps of minimal surfaces in the Euclidean 4-space and their generalizations.

Free Research Field

微分幾何学, 曲面論

Academic Significance and Societal Importance of the Research Achievements

4次元空間内の零平均曲率ベクトルを持つ空間的または時間的曲面の等方性についてのまとまった理解を得ることができ, またWillmore曲面上の正則4次微分および共形Gauss写像の理解が大いに進んだ.
4次元Euclid空間内の極小曲面のGauss写像の正則性は良く知られている. この結果を一般化でき, また空間の種類をLorentzやニュートラルとしても類似の結果および一般化が得られた.