Global properties of minimal surfaces in Euclidean space and zero mean curvature surfaces in Minkowski space

• Project/Area Number: 17K05219
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Hiroshima University (2019-2021); Okayama University (2017-2018)
• Principal Investigator: Fujimori Shoichi 広島大学, 先進理工系科学研究科(理), 教授 (00452706)
• 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: 極小曲面 / 極大曲面 / 平均曲率0曲面 / ワイエルシュトラス型表現公式 / 解析的延長 / 特異点 / 退化極限 / ワイエルストラス表現公式

Outline of Final Research Achievements

The global properties of minimal surfaces in Euclidean space and zero mean curvature surfaces in Minkowski space were investigated. For periodic minimal surfaces in Euclidean 3-space, some new families were constructed and their limits were studied. For zero mean curvature surfaces in Minkowski 3-space, periodic surfaces and nonorientable surfaces were constructed. Moreover, analytic extensions for surfaces which possess Weierstrass type representation formulae were observed.

Academic Significance and Societal Importance of the Research Achievements

ユークリッド空間の周期的極小曲面は界面活性剤膜の数学的モデルであることが知られており、数学者だけでなく物理学者や化学者にとっても重要な研究テーマである。本研究では主に複素解析的手法を用いて周期的極小曲面の研究を行ったが、得られた結果は物理や化学の分野でも応用されることが期待される。一方、ミンコフスキー空間の平均曲率0曲面も同様の手法で研究を行ったが、こちらは特異点が現れるので、特異点論の発展にも寄与していると思われる。

Research Products

• [Int'l Joint Research] Indiana University South Bend/University of Notre Dame(米国)
• [Int'l Joint Research] Korea University(韓国)
• [Int'l Joint Research] Indiana University South Bend/University of Notre Dame(米国)
• [Int'l Joint Research] Korea University(韓国)
• [Int'l Joint Research] インディアナ大学サウスベンド(米国)
• [Int'l Joint Research] 高麗大学校(韓国)
• [Int'l Joint Research] インディアナ大学サウスベンド(米国)
• [Int'l Joint Research] 高麗大学校(韓国)
• [Int'l Joint Research] ウィーン工科大学(オーストリア)
• [Int'l Joint Research] 高麗大学校(韓国)
• [Int'l Joint Research] インディアナ大学(米国)
• [Journal Article] Higher genus nonorientable maximal surfaces in the Lorentz-Minkowski 3-space (2022)
  • Author(s): Shoichi Fujimori and Shin Kaneda
  • Journal Title: Tohoku Mathematical Journal Volume: -
  • Peer Reviewed / Open Access
• [Journal Article] Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space (2020)
  • Author(s): Shoichi Fujimori, Yu Kawakami, Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, Kotaro Yamada
  • Journal Title: Mathematical Journal of Okayama University Volume: 62 Pages: 179-195
  • NAID: 120006778830
  • Peer Reviewed / Open Access
• [Journal Article] On limits of triply periodic minimal surfaces (2018)
  • Author(s): Ejiri Norio、Fujimori Shoichi、Shoda Toshihiro
  • Journal Title: Annali di Matematica Pura ed Applicata (1923 -) Volume: 197 Issue: 6 Pages: 1739-1748
  • DOI: 10.1007/s10231-018-0746-8
  • Peer Reviewed
• [Journal Article] A construction of a two-parameter family of triply periodic minimal surfaces (2018)
  • Author(s): Norio Ejiri, Shoichi Fujimori, Toshihiro Shoda
  • Journal Title: Kobe Journal of Mathematics Volume: 35 Pages: 45-83
  • Peer Reviewed
• [Journal Article] Triply periodic zero mean curvature surfaces in Lorentz-Minkowski 3-space (2018)
  • Author(s): Shoichi Fujimori
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 78 Pages: 201-219
  • Peer Reviewed
• [Journal Article] Quadrics and Scherk towers (2017)
  • Author(s): Fujimori S.、Hertrich-Jeromin U.、Kokubu M.、Umehara M.、Yamada K.
  • Journal Title: Monatshefte f?r Mathematik Volume: - Issue: 2 Pages: 249-279
  • DOI: 10.1007/s00605-017-1075-5
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space (2017)
  • Author(s): Shoichi Fujimori, Yu Kawakami, Masatoshi Kokubu, Wayne Rossman, Masaaki Umehara, and Kotaro Yamada
  • Journal Title: Osaka Journal of Mathematics Volume: 54 Pages: 249-272
  • NAID: 120006312701
  • Peer Reviewed
• [Presentation] 高種数の向き付け不可能な極大曲面について (2022)
  • Author(s): 藤森祥一, 金田伸
  • Organizer: 日本数学会年会
• [Presentation] Zero mean curvature surfaces in Euclidean and Lorentz-Minkowski 3-space, I (2020)
  • Author(s): Shoichi Fujimori
  • Organizer: Discussion meeting on zero mean curvature surfaces
  • Int'l Joint Research / Invited
• [Presentation] Zero mean curvature surfaces in Euclidean and Lorentz-Minkowski 3-space, II (2020)
  • Author(s): Shoichi Fujimori
  • Organizer: Discussion meeting on zero mean curvature surfaces
  • Int'l Joint Research / Invited
• [Presentation] Zero mean curvature surfaces in Euclidean and Lorentz-Minkowski 3-space, III (2020)
  • Author(s): Shoichi Fujimori
  • Organizer: Discussion meeting on zero mean curvature surfaces
  • Int'l Joint Research / Invited
• [Presentation] Constant curvature tori of Joachimsthal type (2020)
  • Author(s): Shoichi Fujimori
  • Organizer: Geometric shape generation
  • Int'l Joint Research / Invited
• [Presentation] Deformations of periodic minimal surfaces and their limits (2019)
  • Author(s): Shoichi Fujimori
  • Organizer: 2019 Langenhop Lecture and SIU Pure Mathematics Conference
  • Int'l Joint Research / Invited
• [Presentation] Limits of periodic minimal surfaces (2019)
  • Author(s): Shoichi Fujimori
  • Organizer: Branched Coverings, Degenerations, and Related Topics 2019
  • Int'l Joint Research / Invited
• [Presentation] 3次元Lorentz-Minkowski空間の平均曲率0曲面について (2018)
  • Author(s): 藤森祥一
  • Organizer: 合宿セミナー 2018 in 福山
  • Invited
• [Presentation] Quadrics and their Christoffel duals (2017)
  • Author(s): Shoichi Fujimori
  • Organizer: International Workshop on Differential Geometric Aspects of Integrable Systems
  • Int'l Joint Research / Invited
• [Presentation] Zero mean curvature surfaces of mixed causal type in the Lorentz-Minkowski 3-space (2017)
  • Author(s): Shoichi Fujimori
  • Organizer: The third Japanese-Spanish workshop on Differential Geometry
  • Int'l Joint Research / Invited
• [Book] 極小曲面論入門 (2019)
  • Author(s): 川上裕, 藤森祥一
  • Total Pages: 111
  • Publisher: サイエンス社
• [Remarks] 藤森祥一のホームページ
  • URL: https://home.hiroshima-u.ac.jp/fujimori/index-j.html
• [Remarks] 藤森祥一のウェブページ
  • URL: https://home.hiroshima-u.ac.jp/fujimori/index-j.html
• [Remarks] 藤森祥一のホームぺージ
  • URL: http://www.math.okayama-u.ac.jp/~fujimori/index-j.html
• [Remarks] 藤森祥一のホームページ
  • URL: http://www.math.okayama-u.ac.jp/~fujimori/index-j.html