2023 Fiscal Year Final Research Report
Study of the moduli space of periodic minimal surfaces by means of geometric invariants
| Project/Area Number |
20K03616
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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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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kansai University (2021-2023)
Saga University (2020) |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 三重周期的な極小曲面 / Morse指数 / 退化次数 / 符号数 / 安定性 |
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| Outline of Final Research Achievements |
A triply periodic minimal surface is a mathematical model of surfactant, and our main theme in this period is to study the moduli space of triply periodic minimal surfaces by means of three kinds of geometric invariants, namely, the Morse index, the nullity, the signature.
We first focus on the genus three case, and computed the three quantities for the mPCLP/mDCLP family, which is a three parameter family and which have been studied in physics in 1990s. Next, we consider a boundary of the moduli space via the Bolza surface which is the genus two classical closed surface. We determined the three quantities around the boundary, and find a local structure of the boundary of the moduli space.
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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 |
界面活性剤という実際に現実の世界に存在する膜が何故その形状をとるのかという疑問が常にある。先述の通り、三重周期的な極小曲面は界面活性剤の数学的モデルである。一方、微分幾何学においては自然現象はある特定の幾何学的不変量の値をとる場合に該当すると考えられている。このことから、幾何学的不変量を計算することによって三重周期的な極小曲面全体の集合の構造を解明することは、自然現象の原理を解明するという学術的および社会的な意義がある。
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