2020 Fiscal Year Research-status Report
Development of analysis and discretization in differential geometry
| Project/Area Number |
20K03585
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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) |
安本 真士 大阪市立大学, 数学研究所, 特別研究員 (70770543)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 離散的微分幾何学 / 離散曲面 / 離散曲線 / 特異点 / Darboux変換 |
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| Outline of Annual Research Achievements |
Our purpose: 1) use transformation theory to discretize equations and surfaces in the smooth category, while preserving underlying structures; 2) consider behavior of surfaces within integrable systems having singularities and signature changes.
An example of transformation theory giving insight into discretization is isothermic surfaces, with their Darboux transforms. By Bianchi permutability, a mesh of Darboux transforms produces a discrete isothermic surface, elucidating these surfaces' definition.
With similar concepts, jointly with J. Cho and T. Seno, we have produced discretization of the potential mKdV equation. Further, jointly with Japanese researchers (see below), singularities, signature changes and analytic extendability of various catenoids in de Sitter 3-space were studied.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Results obtained:
1) With Cho and Seno, we found the semi-discrete potential mKdV equation via Darboux transforms of discrete plane curves preserving arc-length polarization, employing Bianchi permutability, and used this to determine geometric properties of the equation's solutions.
2) Again with Cho and Seno, we also extended these methods to the fully discrete potential mKdV equation, again finding geometric properties.
3) With S. Fujimori, Y. Kawakami, M. Kokubu, M. Umehara and K. Yamada, we studied analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space, using new notions of analytic completeness and arc-properness of images of real analytic maps. We found criteria for analytic completeness, applying this to analytic completeness of those catenoids.
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| Strategy for Future Research Activity |
We will progress toward these objectives:
1) With Fujimori, Kawakami, Kokubu, Umehara, Yamada and S.-D. Yang, we are now studying criteria for unextendability of images of real analytic maps, which will extend our previous work to allow singularities other than cone-like ones, allowing us to evaluate the analytic completeness of Weierstrass-representation-equipped catenoids rather than geometric catenoids.
2) With S. Akamine, M. Yasumoto and Cho, we will examine duality between discrete minimal surfaces in Euclidean 3-space and discrete maximal surfaces in Minkowski 3-space, and establish criteria for fold and cone-point singularities on these surfaces.
3) With Cho and M. Hara, we will develop global closing conditions for surfaces with non-trivial topology, using Moebius geometry.
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| Causes of Carryover |
新型コロナウイルスの影響で予定していた外国と国内旅費が使えなくなった。次年度の旅費に利用する予定です。
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