2022 Fiscal Year Annual Research Report
Differential geometric approach to discrete surfaces
| Project/Area Number |
15K04845
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| Research Institution | Kobe University |
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Principal Investigator |
Rossman W.F 神戸大学, 理学研究科, 教授 (50284485)
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| Co-Investigator(Kenkyū-buntansha) |
直川 耕祐 神戸大学, 理学研究科, 特別研究員(PD) (60740826) [Withdrawn]
佐治 健太郎 神戸大学, 理学研究科, 教授 (70451432)
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| Project Period (FY) |
2015-04-01 – 2023-03-31
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| Keywords | Differential Geometry / Discrete Surface Theory / Lorentzian spaces / Omega surfaces / Darboux transformations / Lie sphere geometry |
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| Outline of Annual Research Achievements |
This was research aimed at extending the rich mathematical structure of smooth surfaces to discretized surfaces. The general goal was to study how singular behavior of various types occurs within integrable systems related to surface geometry, but we also examined new methods for constructing discrete surfaces, including an analysis of the properties that appear on those surfaces. Additionally, we considered smooth surfaces with integrable systems properties, i.e. Weierstrass representations, that can have either singularities or signature type change, or possibly both. More specifically, the following results were obtained:
(1) Working with F. Burstall, U. Hertrich-Jeromin, J. Cho and M. Pember, we developed a theory for discretization of Omega surfaces, including a full transformation theory for such surfaces.
(2) Working with S. Fujimori, Y. Kawakami, M. Kokubu, M. Umehara and K. Yamada, we investigated properties of analytic extensions of constant mean curvature surfaces in Lorentzian spaces, including de Sitter spaces.
(3) Working with J. Cho and T. Seno, we used transformation theory to create a new approach to determining both the semi-discrete and fully discrete potential mKdV equations.
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