Research on trivalent constant mean curvature surfaces from various viewpoints

• Project Area: Discrete Geometric Analysis for Materials Design
• Project/Area Number: 18H04489
• Research Category: Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
• Allocation Type: Single-year Grants
• Research Institution: Kyushu University (2019); Osaka City University (2018)
• Principal Investigator: 安本 真士 九州大学, マス・フォア・インダストリ研究所, 学術研究員 (70770543)
• Project Period (FY): 2018-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000); Fiscal Year 2019: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2018: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
• Keywords: 離散微分幾何 / 極小曲面 / 平均曲率一定曲面 / 可積分系 / 曲面論 / 空間グラフ / 離散微分幾何学 / 三価グラフ

Outline of Annual Research Achievements

2019年度および新型コロナウイルス感染症拡大の影響を受けて繰り越した2020年度は主に以下の成果を得た．
[1] 初年度に得られた，3次元ミンコフスキー空間内の次数3の空間グラフの極大曲面に対する頂点法方向の概念が他の離散化された曲面にも有用であることを確認するため，同空間内の半離散極大曲面にも適用した．当初の予想通り，半離散極大曲面の曲率だけでなく，現れる特異点の解析にも非常に有用であることを示した．この成果は単著論文として国際誌に掲載が決定している．
[2] 離散曲面に対する調和写像の理論をより理解することを目的とし，まずは3次元ミンコフスキー空間内の離散時間的極小曲面の研究に取り組んだ．連続的な場合，時間的平均曲率一定曲面は双曲型偏微分方程式と密接に関わりを持っているため，従来の平均曲率一定曲面の離散化よりも相性が良いと期待される．まずは離散時間的極小曲面に対応するガウス写像にローレンツ調和写像の性質が現れることを示し，さらに座標の取り方によらず，離散時間的極小曲面の各座標関数が離散波動方程式の解となることを示した．さらに，離散曲面に対する実解析的延長問題について取り組み，3次元ミンコフスキー空間内の離散平均曲率零曲面の構成，およびその3次元ユークリッド空間内の離散極小曲面の構成への応用について取り組み始めた．
これらに加えて，数件の国際研究集会およびスクールを開催し，本研究課題に係る研究普及に努めた．特に，対面・オンライン併用で開催した国際研究集会「可視化の数理と，対称性およびモジュライの深化」は参加登録者数約150名の大規模なものとなり，本研究課題の総括として相応しいものとなった．

Research Progress Status

令和元年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和元年度が最終年度であるため、記入しない。

Research Products

• [Int'l Joint Research] トリノ工科大学(イタリア)
• [Int'l Joint Research] ウィーン工科大学(オーストリア)
• [Int'l Joint Research] ブラウン大学(米国)
• [Int'l Joint Research] ウィーン工科大学(オーストリア)
• [Journal Article] Semi-discrete maximal surfaces with singularities in Minkowski space (2021)
  • Author(s): Masashi Yasumoto
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 未定
  • Peer Reviewed
• [Journal Article] 離散曲面の微分幾何 (2021)
  • Author(s): Wayne Rossman, 安本真士
  • Journal Title: 数学 Volume: 73 Pages: 1-21
  • NAID: 40022471610
  • Peer Reviewed
• [Journal Article] SEMI-DISCRETE LINEAR WEINGARTEN SURFACES WITH WEIERSTRASS-TYPE REPRESENTATIONS AND THEIR SINGULARITIES (2020)
  • Author(s): Wayne Rossman, Masashi Yasumoto
  • Journal Title: Osaka Journal of Mathematics Volume: 57 Issue: 1 Pages: 169-185
  • DOI: 10.18910/73744
  • NAID: 120006781495
  • ISSN: 00306126
  • Peer Reviewed
• [Journal Article] A first step to two topics in discretizations of surfaces in Euclidean space (2020)
  • Author(s): Masashi Yasumoto
  • Journal Title: Anam Lecture Notes in Mathematics Volume: 2 Pages: 1-35
• [Journal Article] 3次元ミンコフスキー空間内の離散空間的平均曲率一定曲面 (2020)
  • Author(s): 安本真士
  • Journal Title: 数理解析研究所講究録 Volume: 2152 Pages: 1-15
• [Journal Article] Trivalent maximal surfaces in Minkowski space (2018)
  • Author(s): W.Y. Lam, M. Yasumoto
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 211 Pages: 169-184
  • DOI: 10.1007/978-3-319-66290-9_10
  • ISBN: 9783319662893, 9783319662909
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Discrete linear Weingarten surfaces and their singularities in Riemannian and Lorentzian spaceforms (2018)
  • Author(s): W. Rossman, M. Yasumoto
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 78 Pages: 383-410
  • DOI: 10.2969/aspm/07810383
  • Peer Reviewed
• [Journal Article] Weierstrass-type representations for timelike surfaces (2018)
  • Author(s): M. Yasumoto
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 78 Pages: 449-469
  • DOI: 10.2969/aspm/07810449
  • Peer Reviewed
• [Journal Article] Semi-discrete linear Weingarten surfaces and their singularities in Riemannian and Lorentzian spaceforms (2018)
  • Author(s): W. Rossman, M. Yasumoto
  • Journal Title: Osaka Journal of Mathematics Volume: to appear
  • Peer Reviewed
• [Journal Article] 4色定理の証明 -小さな誤りから大きな問題への道- (2018)
  • Author(s): W. Rossman, 安本真士
  • Journal Title: 数学セミナー Volume: 2018年9月号 Pages: 43-47
  • NAID: 40021656732
• [Journal Article] 3次元ミンコフスキー空間内の離散空間的平均曲率一定曲面 (2018)
  • Author(s): 安本真士
  • Journal Title: RIMS Kokyuroku Volume: to appear
• [Presentation] 離散曲面の微分幾何 (2021)
  • Author(s): 安本真士
  • Organizer: プレ・マス・フォア・イノベーションカフェ
• [Presentation] 可積分変換による離散曲面の構成法 (2021)
  • Author(s): 安本真士
  • Organizer: ワークショップ「幾何学と様々な自然現象の解析」
  • Invited
• [Presentation] Discrete timelike minimal surfaces: two Weierstrass-type representations from discrete wave equations (2020)
  • Author(s): Masashi Yasumoto
  • Organizer: The closing workshop of the project "Geometric Shape Generation"
  • Int'l Joint Research / Invited
• [Presentation] 離散Weierstrass型の表現公式 (2020)
  • Author(s): 安本真士
  • Organizer: 日本数学会2020年度年会
• [Presentation] 離散Weierstrass型の表現公式 (2020)
  • Author(s): 安本真士
  • Organizer: 日本応用数理学会2020年度年会
• [Presentation] 離散化された線形ワインガルテン曲面とその変形族の研究 (2019)
  • Author(s): 安本真士
  • Organizer: 2018年度数学研究会特別賞受賞式・受賞講演
  • Invited
• [Presentation] Discrete constant mean curvature surfaces -Discretization of soap bubbles- (2019)
  • Author(s): Masashi Yasumoto
  • Organizer: Geometric shape generation: integrability, variational analysis and applications
  • Int'l Joint Research / Invited
• [Presentation] Global behavior of discrete surfaces (2019)
  • Author(s): Joseph Cho, Wayne Rossman, Seong-Deog Yang, Masashi Yasumoto
  • Organizer: Geometric shape generation: integrability, variational analysis and applications
  • Int'l Joint Research / Invited
• [Presentation] Discrete zero mean curvature surfaces (2019)
  • Author(s): Masashi Yasumoto
  • Organizer: Discrete and Computational Geometry and its Applications
  • Int'l Joint Research / Invited
• [Presentation] 離散正則関数を用いた離散平均曲率0曲面の構成 (2019)
  • Author(s): 安本真士
  • Organizer: 新学術領域「次世代物質探索のための離散幾何学」研究成果発表会・領域会議
• [Presentation] Discrete Weierstrass-type representations (2019)
  • Author(s): Masashi Yasumoto
  • Organizer: m:iv Autumn 2019 Workshop
  • Int'l Joint Research / Invited
• [Presentation] Discrete Weierstrass-type representations (2019)
  • Author(s): Masashi Yasumoto
  • Organizer: Geometry and Topology seminar, University of Luxembourg
  • Invited
• [Presentation] An invitation to discretization of surfaces from an integrable systems viewpoint (2019)
  • Author(s): M. Yasumoto
  • Organizer: Introductory workshop on discrete differential geometry
  • Int'l Joint Research / Invited
• [Presentation] 3次元ミンコフスキー空間内の離散空間的平均曲率一定曲面 (2018)
  • Author(s): 安本真士
  • Organizer: RIMS共同研究(公開型) 部分多様体の幾何学の深化と展開
  • Invited
• [Presentation] Discrete timelike minimal surfaces (2018)
  • Author(s): M. Yasumoto
  • Organizer: 3rd m:iv workshop
  • Int'l Joint Research / Invited
• [Presentation] Discretization of linear Weingarten surfaces from a viewpoint of integrable systems (2018)
  • Author(s): M. Yasumoto
  • Organizer: Summer School 2018 in Fukuoka "Geometric shape generation''
  • Int'l Joint Research / Invited
• [Presentation] 3次元ミンコフスキー空間内の離散空間的平均曲率一定曲面の構成法 (2018)
  • Author(s): 安本真士
  • Organizer: 広島幾何学研究集会 2018
  • Invited
• [Presentation] Semi-discrete maximal surfaces revisit (2018)
  • Author(s): M. Yasumoto
  • Organizer: The 12th GEOSOCK Seminar "Geometry of Discrete Surfaces and Applications"
  • Invited
• [Presentation] A geometric solution of the discrete sinh-Gordon equation and discrete spacelike constant mean curvature surfaces in Minkowski space (2018)
  • Author(s): M. Yasumoto
  • Organizer: Symmetries and Integrability in Difference Equations (SIDE 13)
  • Int'l Joint Research
• [Funded Workshop] The 27th Osaka City University International Academic Symposium "Mathematical Science of Visualization and Deepening of Symmetry and Moduli" (2021)
• [Funded Workshop] Workshop and School on Geometric Analysis and Discrete Geometry (2020)
• [Funded Workshop] Mini-Workshop on Differential Geometry and its Discretizations (2019)
• [Funded Workshop] The 3rd International Workshop "Geometry of Submanifolds and Integrable Systems" (2019)
• [Funded Workshop] The 2nd International Conference "Geometry of Submanifolds and Integrable Systems" (2019)
• [Funded Workshop] The 13th GEOSOCK Seminar "Recent Progress on Willmore Surfaces" (2019)
• [Funded Workshop] Mini-Workshop on Geometry and Mathematical Science (2018)
• [Funded Workshop] The 12th GEOSOCK Seminar "Geometry of Discrete Surfaces and Applications" (2018)