2021 Fiscal Year Final Research Report
The representation formulas for a surface of higher codimension and a submanifold and their application
| Project/Area Number |
17K05217
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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 | University of Hyogo (2019-2021)
University of Tsukuba (2017-2018) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
長谷川 和志 金沢大学, 学校教育系, 教授 (50349825)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | 変換 / 極小曲面 / 共形写像 / 調和写像 / ウィルモア曲面 |
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| Outline of Final Research Achievements |
I studied how to construct a concrete example of a surface. We obtained a new method for constructing a new one from a given one for a surface which is a mathematical object corresponding to a soap film (minimal surface) in a four-dimensional Euclidean space. We have extended a representation formula and transforms that were known about general surfaces in 4-dimensional Euclidean space to surfaces in n-dimensional Euclidean space. We extended the transforms known for the minimal surface in the 3-sphere to the transforms of surfaces in the n-sphere, and found that if a given surface is minimal, then the transform is the transforms between minimal surfaces.
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| Free Research Field |
微分幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
一般に微分方程式の解を全て求めることは難しい. 代わりに, 解が一つ与えられているときに, それを用いて新たな解を構成できることがある. その理論を変換の理論という. 変換の理論が作れるのは, 関連する数学が十分発展している微分方程式である.極小曲面の微分方程式はそのような微分方程式のうちの一つである.本研究では, すでに知られていた極小曲面の微分方程式の変換の理論を, 新たな計算方法を導入して, 次元が高い場合に拡張した. これにより, すでに知られていた変換の新たな意味づけや, 変換の理論の構成の可能性がある微分方程式の候補が得られた.
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