2023 Fiscal Year Research-status Report
Global behavior of discrete surfaces via integrability
| Project/Area Number |
23K03091
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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) |
佐治 健太郎 神戸大学, 理学研究科, 教授 (70451432)
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| Project Period (FY) |
2023-04-01 – 2028-03-31
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| Keywords | 離散的微分幾何学 / 曲面理論 / 可積分系 / 特異点 / Darboux変換 |
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| Outline of Annual Research Achievements |
The purpose of this research is to develop the connection between complex analytic methods in surface theory with the integrable systems methods in transformation theory, and produce new results with this.
The former methods involve primarily the use of Weierstrass and DPW type representations to construct surfaces with particular curvature properties, the first example of this being minimal surfaces in Euclidean space, but including many other classes of surfaces in a variety of spaceforms. The latter methods include transformations of surfaces, such as Baecklund and Darboux transformations, together with permutability properties, in a Moebius geometric context.
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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
1) Together with J. Cho, M. Pember, F. Burstall and U. Hertrich-Jeromin, we have unified numerous descriptions of discrete Omega surfaces, and have extended the notions of their transformations, including determining Darboux transforms for all such surfaces.
2) Together with S. Fujimori, M. Kokubu, Y. Kawakami, M. Umehara, K. Yamada and S.-D. Yang, we have considered analytic extensions of surfaces, with applications to particular types of surfaces in Lorentzian space such as Minkowski 3-space and de Sitter 3-space, and especially understanding how maximal surfaces in Minkowski 3-space can extend in various ways (possibly becoming timelike in the extensions) and understanding all ways that the class of constant mean curvature 1 catenoids in de Sitter 3-space extend.
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| Strategy for Future Research Activity |
1) Together with T. Raujouan and N. Suda, we will consider Darboux transformations of discrete constant Gaussian curvature surfaces of revolution, extending previous work by T. Hoffmann and A. Sagemann-Furnas, and thereby creating families of new non-rotational examples of such surfaces.
2) Together with J. Cho, M. Hara and T. Raujouan, we will apply Darboux transforms of holomorphic functions in the plane to producing surfaces by inserting these functions into Weierstrass representations, creating new examples in a number of spaceforms. We will also give general results about their end behavior and singularity behavior.
3) Together with K. Leschke, F. Pedit and F. Burstall, we will consider how to produce discrete and semidiscrete isothermic tori that are full in higher dimensional Euclidean spaces.
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| Causes of Carryover |
2023年度前半に入院加療をした為、外国出張に使用できなかった。次年度の旅費に利用する予定です。
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