| 研究課題/領域番号 |
23K03091
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分11020:幾何学関連
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| 研究機関 | 神戸大学 |
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研究代表者 |
Rossman W.F 神戸大学, 理学研究科, 教授 (50284485)
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| 研究分担者 |
佐治 健太郎 神戸大学, 理学研究科, 教授 (70451432)
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| 研究期間 (年度) |
2023-04-01 – 2028-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,550千円 (直接経費: 3,500千円、間接経費: 1,050千円)
2027年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2026年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2025年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2024年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2023年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
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| キーワード | 離散的微分幾何学 / 曲面理論 / 可積分系 / 特異点 / Darboux変換 / differential geometry / discrete surface theory / transformation theory / integrable structure / global behavior |
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| 研究開始時の研究の概要 |
The overall outline of this research is to use the transformation theory to provide a framework that offers a natural way to discretize surfaces, by employing permutability properties of transforms. In fact, transformation theory itself can be seen in a unified context with surface discretization.
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| 研究実績の概要 |
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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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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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| 今後の研究の推進方策 |
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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