| Project/Area Number |
15K04845
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kobe University |
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Principal Investigator |
Rossman W.F 神戸大学, 理学研究科, 教授 (50284485)
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| Co-Investigator(Kenkyū-buntansha) |
直川 耕祐 神戸大学, 理学研究科, 特別研究員(PD) (60740826)
佐治 健太郎 神戸大学, 理学研究科, 准教授 (70451432)
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| Project Period (FY) |
2015-04-01 – 2022-03-31
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| Project Status |
Granted (Fiscal Year 2020)
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| Budget Amount *help |
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 離散的微分幾何学 / 離散曲面 / 離散曲線 / 特異点 / Darboux変換 / 半離散曲面 / surface theory / discrete geometry / minimal surface / Omega surface / transformation theory / Lie sphere geometry / channel surfaces / orthogonal systems / Guichard surfaces / Omega surfaces / discrete surfaces / semi-discrete surfaces / integrable systems / Moebius geometry |
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| Outline of Annual Research Achievements |
This research considers surfaces with integrable systems properties that can have singularities and/or signature type change, giving understanding of behavior at such points, and demonstrates the relationships between singularities on the one hand and the points of signature change on the other. In joint work with a number of Japanese mathematicians as listed below, we have understood singularities and signature changes and analytic extendability of constant mean curvature one catenoids of various types in de Sitter 3-space. Additionally, we used transformation theory to create a new approach to determining both the semi-discrete and fully discrete potential mKdV equations, in joint work with J. Cho and T. Seno, analogously to how Darboux transforms produce discrete isothermic surfaces.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
With S. Fujimori, Y. Kawakami, M. Kokubu, M. Umehara and K. Yamada, we considered the concept of analytic extensions of Sitter 3-space geometric catenoids with constant mean curvature one, which required new notions of analytic completeness and arc-properness of images of real analytic maps. The notion of arc-properness in particular was quite helpful for understanding the extensions in relation to singular behaviors (in particular, cone-like points). We then developed useful criteria for analytic completeness, and as an application, used this to determine analytic completeness of those geometric catenoids, and also discovered how one can similarly approach a different class of catenoids built up from a more algebraic perspective, that is, catenoids with Weierstrass representations.
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| Strategy for Future Research Activity |
We have the following objectives:
1) With S. Akamine, M. Yasumoto and Cho, we will examine duality between discrete minimal surfaces in Euclidean 3-space and discrete maximal surfaces in Minkowski 3-space, and establish criteria for fold and cone-point singularities on these surfaces. Also, With Cho and M. Hara, we will develop global closing conditions for surfaces with non-trivial topology, using Moebius geometry.
2) With Fujimori, Kawakami, Kokubu, Umehara, Yamada and S.-D. Yang, we are now studying criteria for unextendability of images of real analytic maps, which will extend our previous work to allow singularities other than cone-like ones, allowing us to evaluate the analytic completeness of Weierstrass-representation-equipped catenoids rather than geometric catenoids.
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