| 研究課題/領域番号 |
15K04845
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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 |
直川 耕祐 神戸大学, 理学研究科, 特別研究員(PD) (60740826) [辞退]
佐治 健太郎 神戸大学, 理学研究科, 准教授 (70451432)
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| 研究期間 (年度) |
2015-04-01 – 2022-03-31
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| キーワード | 離散的微分幾何学 / 離散曲面 / 離散曲線 / 特異点 / Darboux変換 |
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| 研究実績の概要 |
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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| 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
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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| 今後の研究の推進方策 |
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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| 次年度使用額が生じた理由 |
新型コロナウイルスの影響で予定していた外国と国内旅費が使えなくなった。次年度の旅費に利用する予定です。
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