2019 Fiscal Year Final Research Report
Study of stability of periodic minimal surfaces and their limits
| Project/Area Number |
16K05134
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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 | Saga University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 三重周期的な極小曲面 / 体積保存安定性 |
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| Outline of Final Research Achievements |
Periodic minimal surfaces in the Euclidean three-space can be considered as a mathematical model for surfactant in the soft matter. In 1990s, physicists considered many families of triply periodic minimal surfaces. On the other hand, a stability of a minimal surface has been studied via area minimizing situation. In particular, Barbosa-doCarmo's technique related to volume preserving stability might be useful tool for this situation. By this study, we find out volume preserving stability for some families which physicists considered. Moreover, we gave mathematical description of lamellar phases in the soft matter as limits of triply periodic minimal surfaces.
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| Free Research Field |
極小曲面
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| Academic Significance and Societal Importance of the Research Achievements |
界面活性剤の膜の変化の仕方・規則性を数学的に記述することにより，自然現象を解明するというのが本研究課題の意図である．膜は一時的な変化の後に安定した状態になる．この安定した状態が体積保存安定性によって記述できると考え，物理学者たちが考察してきた変形族の体積保存安定性を特定したというのが本研究である．また，界面活性剤の膜の温度を変化させた際に起こる膜の変異が三重周期的な極小曲面の極限として記述されると考え，特殊な場合の極限を特定することによりその変異に類似した対象を得た．これにより，膜の変異の理論的必然性の示唆が得られたと考える．
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