Principal distributions on surfaces in various spaces

• 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
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• 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 / Minkowski空間 / 曲面 / リフト / 正則 / 2重外積空間 / 光錐 / ニュートラル超Kaehler / 4次元Lorentz多様体 / 平均曲率ベクトルが零 / 混合型構造 / 複素4次微分 / 階数4のベクトル束 / ツイスター空間 / 水平な切断 / (ニュートラル)超Kaehler多様体 / (パラ)複素4次微分 / 共変微分が光的 / Willmore型の時間的曲面 / Gauss-Codazzi-Ricciの方程式 / ニュートラル多様体 / 等方的 / 複素曲線 / Willmore型曲面 / Hopf４次微分 / (-k)-写像 / 正則はめこみ / 主分布 / 平均曲率ベクトル / 臍点 / 過剰決定系

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.

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.

Research Products

• [Journal Article] Regularity of ends of zero mean curvature surfaces in \mathbf{R}^{2, 1} (2022)
  • Author(s): Ando Naoya, Hamada Kohei, Hashimoto Kaname, Kato Shin
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 未定
  • Peer Reviewed
• [Journal Article] Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector (2022)
  • Author(s): Ando Naoya
  • Journal Title: Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg Volume: 未定 Issue: 1 Pages: 105-123
  • DOI: 10.1007/s12188-021-00254-y
  • Peer Reviewed
• [Journal Article] ISOTROPICITY OF SURFACES WITH ZERO MEAN CURVATURE VECTOR IN 4-DIMENSIONAL SPACES (2022)
  • Author(s): ANDO Naoya
  • Journal Title: New Horizons in Differential Geometry and its Related Fields Volume: - Pages: 177-191
  • DOI: 10.1142/9789811248108_0011
  • ISBN: 9789811248092, 9789811248108
  • Peer Reviewed
• [Journal Article] Isotropicity of surfaces with zero mean curvature vector in 4-dimensional spaces (2021)
  • Author(s): Naoya Ando
  • Journal Title: New Horizons in Differential Geometry and its Related Fields Volume: -
  • Peer Reviewed
• [Journal Article] Horizontality in the twistor spaces associated with vector bundles of rank 4 on tori (2021)
  • Author(s): Naoya Ando and Takumu Kihara
  • Journal Title: Journal of Geometry Volume: 112 Issue: 2
  • DOI: 10.1007/s00022-021-00583-6
  • Peer Reviewed
• [Journal Article] Surfaces with zero mean curvature vector in neutral 4-manifolds (2020)
  • Author(s): Naoya Ando
  • Journal Title: Differential Geometry and its Applications Volume: 72 Pages: 101647-101647
  • DOI: 10.1016/j.difgeo.2020.101647
  • Peer Reviewed
• [Journal Article] SURFACES IN PSEUDO-RIEMANNIAN SPACE FORMS WITH ZERO MEAN CURVATURE VECTOR (2020)
  • Author(s): Naoya Ando
  • Journal Title: Kodai Math. J. Volume: 43 Pages: 193-219
  • NAID: 130007812084
  • Peer Reviewed
• [Journal Article] Complex Curves and Isotropic Minimal Surfaces in HyperKaehler 4-Manifolds (2019)
  • Author(s): Naoya Ando
  • Journal Title: Recent Topics in Differential Geometry and its Related Fields (Proceedings of the 6th International Colloquium on Differential Geometry and its Related Fields) Volume: - Pages: 45-61
  • Peer Reviewed
• [Journal Article] Complex curves and isotropic minimal surfaces in hyperKaehler 4-manifolds (2019)
  • Author(s): Naoya Ando
  • Journal Title: Recent topics in Differential Geometry and its Related Fields Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Over-determined systems in relation to principal curvatures (2017)
  • Author(s): Naoya Ando
  • Journal Title: Journal of Geometry Volume: 108 Issue: 2 Pages: 355-373
  • DOI: 10.1007/s00022-016-0343-1
  • Peer Reviewed
• [Journal Article] C^1-umbilics with arbitrarily high indices (2017)
  • Author(s): N. Ando, T. Fujiyama and M. Umehara
  • Journal Title: Pasific Journal of Mathematics Volume: 288 Issue: 1 Pages: 1-27
  • DOI: 10.2140/pjm.2017.288.1
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] The lifts of the conformal Gauss maps of minimal surfaces in the Euclidean 3-space (2022)
  • Author(s): 安藤直也
  • Organizer: Workshop on Surface Theory ---UY60---
  • Int'l Joint Research / Invited
• [Presentation] 階数4のベクトル束に付随するツイスター空間およびそれらの類似物の切断 (2021)
  • Author(s): 安藤直也
  • Organizer: 福岡大学微分幾何研究集会
  • Invited
• [Presentation] The lifts of surfaces in neutral 4-manifolds into their 2-Grassmann bundles (2021)
  • Author(s): Naoya Ando
  • Organizer: The X International Meeting on Lorentzian Geometry, GeLoCor2021
  • Int'l Joint Research
• [Presentation] The lifts of surfaces in neutral 4-manifolds into their 2-Grassmann bundles (2021)
  • Author(s): 安藤直也
  • Organizer: 大阪市立大学・幾何学講演会
  • Invited
• [Presentation] Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector (2021)
  • Author(s): 安藤直也
  • Organizer: 大阪市立大学・幾何学講演会
  • Invited
• [Presentation] 様々な4次元空間内の平均曲率ベクトルが零である曲面について (2020)
  • Author(s): 安藤直也
  • Organizer: 九大幾何学セミナー
  • Invited
• [Presentation] 4次元ニュートラル空間内の平均曲率ベクトルが零である曲面 (2019)
  • Author(s): Naoya Ando
  • Organizer: DGA2019
  • Int'l Joint Research
• [Presentation] ４次元ニュートラル空間内の平均曲率ベクトルが零である曲面 (2019)
  • Author(s): 安藤直也
  • Organizer: 福岡大学微分幾何研究集会
• [Presentation] 共形Gauss写像 (2019)
  • Author(s): 安藤直也
  • Organizer: 大阪市立大学微分幾何学セミナー
  • Invited
• [Presentation] 時間的曲面の共形Gauss写像およびツイスター・リフト (2019)
  • Author(s): 安藤直也
  • Organizer: 多様体上の変分問題とその周辺領域
• [Presentation] Holomorphic quartic differentials on surfaces (2018)
  • Author(s): Naoya Ando
  • Organizer: The International Colloquium on Differential Geometry and its Related Fields
  • Int'l Joint Research
• [Presentation] Euclid空間内の極小曲面、Gauss写像および正則４次微分 (Minimal surfaces in Euclidean spaces, the Gauss maps and holomorphic quartic differentials) (2018)
  • Author(s): 安藤直也 (Naoya Ando)
  • Organizer: アクセサリー・パラメーター研究会 (Workshop on Accessory parameters at Kumamoto)
  • Int'l Joint Research / Invited
• [Presentation] Euclid空間内の部分多様体の反転による像のコンパクト化について (2018)
  • Author(s): 安藤直也
  • Organizer: 榎本一之教授退職直前ワークショップ
  • Invited
• [Presentation] 様々な４次元空間内の平均曲率ベクトルが零である空間的曲面 (2017)
  • Author(s): 安藤直也
  • Organizer: RIMS研究集会「部分多様体論の潮流」
  • Invited
• [Presentation] 擬Riemann空間型内の平均曲率ベクトルが零である空間的曲面 (2017)
  • Author(s): 安藤直也
  • Organizer: 2017年度福岡大学微分幾何研究集会
• [Remarks] researchmap 安藤 直也
  • URL: https://researchmap.jp/7000016851
• [Remarks] research map 安藤直也
  • URL: https://researchmap.jp/7000016851
• [Remarks] ORCiD Naoya Ando
  • URL: https://orcid.org/0000-0001-8268-1513
• [Remarks] researchmap安藤直也
  • URL: https://researchmap.jp/7000016851
• [Remarks] Naoya ANDO's HomePage
  • URL: http://www.sci.kumamoto-u.ac.jp/~ando/index.html
• [Remarks] 安藤直也のホームページ
  • URL: http://www.sci.kumamoto-u.ac.jp/~ando/index-j.html