2021 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 |
This research is focused on using transformation theory to guide us in understanding how to discretize smooth objects, still preserving the underlying mathematical structures. In this context, much can be explored, such as the following:
(*) singularity theory, signature changes, discretization of famous equations such as mKdV equation, Darboux transformations and other transformations of discretized objects, permutability of transformations, variational methods applied to discretized objects, and notions of extendability and completeness.
Most recently, this research has been exploring ways to extend global isothermic objects to spaces of higher dimension, in joint work with F. Pedit and K. Leschke.
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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
The work with J. Cho and T. Seno has now been completely finalized.
The work with S. Fujimori, Y. Kawakami, M. Kokubu, M. Umehara, K. Yamada, and S-D. Yang on completeness of geometric catenoids in de Sitter 3-space has come to fruition, and this same group has now made significant progress in the analogous question for catenoids with holomorphic representations.
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| Strategy for Future Research Activity |
Progress is now being made in three newer research topics encompassed within this research field:
1) determining and controlling eigenvalues of the Laplacian on a minimal surface, in joint work with A. Paganini, R. Fernandes and D. Polly,
2) extending isothermic surfaces on non-trivial topology to higher dimensional conformal n-spheres, in joint work with F. Pedit and K. Leschke,
3) extending examples of periodic discrete minimal surfaces in Euclidean space to periodic discrete constant anisotropic mean curvature surfaces, in joint work with M. Yasumoto and Y. Jikumaru, via variational methods.
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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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