A study of systems of differential equations associated with projectively minimal surfaces

• Project/Area Number: 19540080
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Fukui University of Technology
• Principal Investigator: SASAKI Takeshi Fukui University of Technology, 自然科学系先端融合研究環, 名誉教授 (00022682)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 射影曲面 / 超幾何微分方程式 / 射影極小曲面 / アフィン球面 / 双曲シュバルツ写像

Research Abstract

We defined the flat surfaces in the 3-dimensional hyperbolic space by hypergeometric differential equations and studied the singularities of such surfaces, their dependence on the parameters of the equations and the asymptotic behavior of the surfaces by using the computer algebra and a clarification of the singularities of mappings. We proposed also the definitions of discrete flat surfaces by using some discrete integrable systems. We gave surveys on the study of projective minimal surfaces and proposed research problems on transformation of such surfaces in terms of line congruence and on a study of related differential equations.

Research Products

• [Journal Article] The hypergeometric differential equation 3E2 with cubic curves as Schwarz images (2010)
  • Author(s): M. Kato, T. Sasaki
  • Journal Title: to appear in Kyushu J. Math 64
  • NAID: 130000884045
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the hyperbolic Schwarz map at irregular singular points (2010)
  • Author(s): T. Koike, T. Sasaki, Masaaki Yoshida
  • Journal Title: Funkcialaj Ekvacioj (in press)
  • NAID: 130000254361
  • Peer Reviewed
• [Journal Article] Singularities of flat fronts and their caustics, and an example arising from the hyperbolic Schwarz map of a hypergeometric equation (2009)
  • Author(s): T. Sasaki, M. Yoshida
  • Journal Title: Results in Math. 56 Pages: 369-385
  • Peer Reviewed
• [Journal Article] Surface singularities appeared in the hyperbolic Schwarz map for the hypergeometric equation, Lecture Notes of the Summer School "Arrangements, Local Systems and Singularities, " (2009)
  • Author(s): T. Sasaki, M. Yoshida
  • Journal Title: Progress in Math 283 Pages: 247-272
  • Peer Reviewed
• [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation (2009)
  • Author(s): K. Saji, T. Sasaki, M. Yoshida
  • Journal Title: J. Math. Soc. Japan 61 Pages: 559-578
  • NAID: 10024905825
  • Peer Reviewed
• [Journal Article] Surface singularities appeared in the hyperbolic Schwarz map for the hypergeometric equation (2009)
  • Author(s): M.Yoshida
  • Journal Title: Progress in Math. 283 Pages: 247-272
  • Peer Reviewed
• [Journal Article] Singularities of flat fronts and their caustics, and an example arising from the hyperbolic Schwarz map of a hypergeometric equation (2009)
  • Author(s): M.Yoshida
  • Journal Title: Results in Math. 56 Pages: 369-385
  • Peer Reviewed
• [Journal Article] Hyperbolic Schwarz map of the confluent hypergeometric differential equation (2009)
  • Author(s): K.Saji
  • Journal Title: J.Math.Soc.Japan 61 Pages: 559-578
  • Peer Reviewed
• [Journal Article] Hyperbolic Schwarz maps of the Airy and the confluent hypergeometric differential equations and their asymptotic behaviors (2008)
  • Author(s): T. Sasak, M. Yoshida
  • Journal Title: J. Math. Sci. Univ. Tokyo 15 Pages: 195-218
  • Peer Reviewed
• [Journal Article] The hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki, K. Yamada, M. Yoshida
  • Journal Title: Experimental Math. 17 Pages: 269-282
  • Peer Reviewed
• [Journal Article] Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts (2008)
  • Author(s): T. Sasaki, K. Yamada, M. Yoshida
  • Journal Title: International J. Math. 19 Pages: 847-863
  • Peer Reviewed
• [Journal Article] Confluence of swallowtail singularities of the hyperbolic Schwarz map defined by the hypergeometric differential equation (2008)
  • Author(s): M. Noro, T. Sasaki, K. Yamada, M. Yoshida
  • Journal Title: Experimental Math. 17 Pages: 191-204
  • Peer Reviewed
• [Journal Article] Interpolation on Markoff transformations on the Fricke surface (2008)
  • Author(s): T. Sasaki, M. Yoshida
  • Journal Title: Tohoku Math. J. 60 Pages: 23-36
  • NAID: 110006602739
  • Peer Reviewed
• [Journal Article] Homogeneous twomanifolds with an invariant two-form (2008)
  • Author(s): T. Sasaki, M. Yoshida
  • Journal Title: Journal of Geometry 88 Pages: 149-161
  • Peer Reviewed
• [Journal Article] Confluence of swallowtail singularities of the hyperbolic Schwarz map defined by the hypergeometric differential equation (2008)
  • Author(s): M. Noro
  • Journal Title: Experimental Math. 17 Pages: 191-204
  • Peer Reviewed
• [Journal Article] Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts (2008)
  • Author(s): T. Sasaki
  • Journal Title: International J. Math. 19 Pages: 847-863
  • Peer Reviewed
• [Journal Article] The hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki
  • Journal Title: Experimental Math. 17 Pages: 269-282
  • Peer Reviewed
• [Presentation] 射影極小曲面の幾何 (2010)
  • Author(s): 佐々木武
  • Organizer: 複素幾何2010(Mabuchi 60)
  • Place of Presentation: 大阪大学
  • Year and Date: 2010-03-19
• [Presentation] 射影極小曲面の幾何 (2010)
  • Author(s): 佐々木武
  • Organizer: 複素幾何2010(Mabuchi 60)
  • Place of Presentation: 大阪大学(大阪府)
  • Year and Date: 2010-03-19
• [Presentation] The hyperbolic Schwarz map for the Gauss hypergeometric differential equation (2009)
  • Author(s): 佐々木武
  • Organizer: 多変数関数論冬セミナー
  • Place of Presentation: 岐阜大学
  • Year and Date: 2009-12-19
• [Presentation] Hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki
  • Organizer: Geometry and arithmetic around hypergeometric functions
  • Place of Presentation: Math. Forsch. Oberwolfach, Germany
  • Year and Date: 2008-12-02
• [Presentation] Singularities of the hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki
  • Organizer: Colloquium on Differential Geometry
  • Place of Presentation: Univ. Granada, Spain
  • Year and Date: 2008-09-25
• [Presentation] Geometry of the hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki
  • Organizer: Conference on Differential Geometry
  • Place of Presentation: Math. Inst. Bedlewo, Poland.
  • Year and Date: 2008-06-23
• [Presentation] Geometry of the hyperbolic Schwarz map for the hypergeometric differential equation (2008)
  • Author(s): T. Sasaki
  • Organizer: Conference on Differential Geometry
  • Place of Presentation: Math. Inst. Bedlewo
  • Year and Date: 2008-06-23
• [Presentation] 凸領域の幾何-アフィン微分幾何の方法 (2008)
  • Author(s): 佐々木武
  • Organizer: 計算による数理科学の展開
  • Place of Presentation: 神戸大学
  • Year and Date: 2008-01-11
• [Presentation] 射影極小曲面の楽しみ (2008)
  • Author(s): 佐々木武
  • Organizer: 福岡・札幌幾何学セミナー
  • Place of Presentation: 九州大学
• [Presentation] Hyperbolic Schwarz map for the hypergeometric differential equation III, IV (2007)
  • Author(s): T. Sasaki
  • Organizer: Arrangements, Local Systems and Singularities
  • Place of Presentation: Galatasaray University, Turkey.
• [Presentation] Hyperbolic Schwarz map for the hypergeometric differential equation III, IV (2007)
  • Author(s): T. Sasaki
  • Organizer: Attangements, Local Systems and Singularities
  • Place of Presentation: Galatasaray University, Turkey
• [Remarks] ホームページ等
  • URL: http://www.math.kobe-u.ac.jp/~sasaki
• [Remarks]
  • URL: http://www.math.kobe-u.ac.jp/~sasaki