| 研究課題/領域番号 |
23KF0051
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| 研究種目 |
特別研究員奨励費
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| 配分区分 | 基金 |
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| 応募区分 | 外国 |
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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 |
RAUJOUAN THOMAS 神戸大学, 理学研究科, 外国人特別研究員
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| 研究期間 (年度) |
2023-04-25 – 2025-03-31
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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 |
2,000千円 (直接経費: 2,000千円)
2024年度: 1,000千円 (直接経費: 1,000千円)
2023年度: 1,000千円 (直接経費: 1,000千円)
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| キーワード | 曲面理論 / 可積分系 / Darboux変換 / DPW方法 |
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| 研究開始時の研究の概要 |
The DPW method will be the primary tool in this research, both for smooth surfaces and for discrete surfaces. Topological questions will be considered, and applied to create surfaces with nontrivial topologies. For such complicated surfaces, symmetry properties and embeddedness will then be considered, as described below.
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| 研究実績の概要 |
We aim at constructing examples of surfaces with constant mean curvature in various ambient spaces using integrable system techniques in the context of holomorphic maps. These techniques find their origins in the Weierstrass representation (1866), which have been extended by DPW (1998) and are now one of the main tools for this task. They extend to the construction of discrete analogues of smooth 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)
1: 当初の計画以上に進展している
理由
We have investigated Delaunay ends for constant mean curvature (CMC) surfaces in Euclidean and hyperbolic space when constructed via the DPW method. We developed a method to check wether a surface arising from DPW has self-intersections, and constructed new examples of complete, embedded, CMC surfaces with any number of Delaunay ends in the hyperbolic space.
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| 今後の研究の推進方策 |
1) With N. Schmitt and J. Ziefle: we have been translating the Weierstrass and Bryant reprensentations for minimal surfaces into a gauge theoretic framework. This allows for the construction of catenoidal ends arising from Fuchsian systems, and a dressing action on the holomorphic frame induces what should be a Darboux transformation.
2) With L. Heller, we are constructing new examples of minimal surfaces in the three-dimensional sphere which have high genus and are not Lawson surfaces. We will obtain surfaces constructed by Kapouleas and should be able to compute their area, as their genus goes to infinity.
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