Study of the moduli theory of periodic minimal surfaces in terms of differential geometry

• Project/Area Number: 24740047
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Saga University
• Principal Investigator: Shoda Toshihiro 佐賀大学, 文化教育学部, 准教授 (10432957)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 幾何学 / 周期的極小曲面 / Morse指数 / 退化次数 / 面積最小 / 極小曲面 / モジュライ理論 / 国際情報交換

Outline of Final Research Achievements

We would study periodic minimal surfaces in the Euclidean space in terms of the differential geometry. In particular, we treated Morse indices of triply periodic minimal surfaces. In this period, we could succeed in computing many Morse indices for families of minimal surfaces which have been studied in physics and chemistry. Also, we could succeed in mathematical description of Lamellar structure by limits of triply perioddic minimal surfaces. Moreover, we obtained the existence and uniqueness results for a comlete minimal surface of finite total curvature in the Euclidean space.

Research Products

• [Journal Article] Minimal surfaces with two ends which have the least total absolute curvature (2016)
  • Author(s): Shoichi Fujimori and Toshihiro Shoda
  • Journal Title: Pacific Journal of Mathematics Volume: 282 Issue: 1 Pages: 107-144
  • DOI: 10.2140/pjm.2016.282.107
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A remark on limits of triply periodic minimal surfaces of genus 3 (2015)
  • Author(s): N. Ejiri, S. Fujimori and T. Shoda
  • Journal Title: Topology and its Applications Volume: 196 Pages: 880-903
  • DOI: 10.1016/j.topol.2015.05.014
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On a moduli theory of minimal surfaces (2014)
  • Author(s): 庄田敏宏，江尻典雄
  • Journal Title: Prospects of Differential Geometry and its related fields Volume: 1 Pages: 155-172
  • Peer Reviewed
• [Journal Article] A family of complete minimal surfaces of finite total curvature with two ends (2014)
  • Author(s): 庄田敏宏，藤森祥一
  • Journal Title: Differential Geometry of submanifolds and its related topics Volume: 1 Pages: 19-31
  • Peer Reviewed
• [Presentation] 極小曲面の存在と一意性について (2016)
  • Author(s): 庄田敏宏
  • Organizer: Aspects of Differential Geometry
  • Place of Presentation: 東北大学理学部
  • Year and Date: 2016-03-09
  • Invited
• [Presentation] 3次元Euclid空間内の全曲率有限な完備極小曲面の存在と一意性について (2016)
  • Author(s): 庄田敏宏
  • Organizer: 2016名城大学幾何学研究集会｢種々の幾何構造の深化｣
  • Place of Presentation: 名城大学
  • Year and Date: 2016-03-01
  • Invited
• [Presentation] 三重周期極小曲面の周期性の崩壊とラメラ構造について (2015)
  • Author(s): 庄田敏宏
  • Organizer: 部分多様体幾何とリー群作用2015
  • Place of Presentation: 東京理科大学森戸記念館
  • Year and Date: 2015-09-07
  • Invited
• [Presentation] On zero eigenfunctions of Jacobi operator on a minimal surface (2014)
  • Author(s): 庄田敏宏
  • Organizer: 4th International Colloquium on Differential Geometry and its Related Fields
  • Place of Presentation: Veliko Tarnovo University (Bulgaria)
  • Year and Date: 2014-09-10
  • Invited
• [Presentation] The existence of non-holomorphic hyperelliptic minimal surfaces in flat 4-tori and their Morse indices (2014)
  • Author(s): 庄田敏宏
  • Organizer: The 5th International Workshop on Differential Geometry and Analysis
  • Place of Presentation: 虹ノ松原ホテル
  • Year and Date: 2014-06-01
  • Invited
• [Presentation] Periodic minimal surfaces in terms of Morse indices
  • Author(s): 庄田敏宏
  • Organizer: Conference on Geometry
  • Place of Presentation: Galatasaray University (Turkey)
  • Invited
• [Presentation] On families of periodic minimal surfaces
  • Author(s): 庄田敏宏
  • Organizer: Progress of geometric structures on manifolds
  • Place of Presentation: 名城大学
  • Invited
• [Presentation] Behavior of one-parameter families of minimal surfaces in flat tori in terms of Morse index
  • Author(s): 庄田敏宏
  • Organizer: The second Japanese-Spanish workshop on Differential Geometry
  • Place of Presentation: 名城大学
  • Invited
• [Presentation] A remark of limits of triply periodic minimal surfaces
  • Author(s): 庄田敏宏
  • Organizer: International Conference on Topology and Geometry 2013
  • Place of Presentation: 島根大学
  • Invited
• [Presentation] Morse indices of minimal surfaces in flat tori
  • Author(s): 庄田敏宏
  • Organizer: Oberwolfach workshop｢Progress in Surface Theory｣
  • Place of Presentation: Oberwolfach研究所(ドイツ)
  • Invited
• [Presentation] On classical triply periodic minimal surfaces
  • Author(s): Toshihiro Shoda
  • Organizer: The eighth Chinese-Japan Friendship Conference on Differential Geometry
  • Place of Presentation: 四川大学
  • Invited
• [Presentation] 極小曲面のMorse index について
  • Author(s): 庄田敏宏
  • Organizer: 日本数学会 2012年度秋季総合分科会 幾何学分科会 特別講演
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] 3 重周期極小曲面の周期計算について
  • Author(s): 庄田敏宏
  • Organizer: 福岡大学幾何学研究会
  • Place of Presentation: 福岡大学
  • Invited
• [Presentation] 4 次元平坦トーラス内の偶数種数の超楕円型極小曲面について
  • Author(s): 庄田敏宏
  • Organizer: 名城大学研究集会「多様体と幾何構造の融合」
  • Place of Presentation: 名城大学
  • Invited
• [Presentation] 物理・化学で扱われている三重周期的極小曲面のMorse index について
  • Author(s): 庄田敏宏
  • Organizer: 若狭三方幾何学研究集会 2013
  • Place of Presentation: 若狭三方
  • Invited
• [Remarks] 庄田敏宏ホームページ「業績リスト」
  • URL: http://extwww.cc.saga-u.ac.jp/~tshoda/shoda-home-j.html