2017 Fiscal Year Annual Research Report
A geometric characterization of discrete Guichard nets and their transformations in integrable geometry
| Project/Area Number |
17F17734
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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) |
SZEWIECZEK GUDRUN 神戸大学, 理学研究科, 外国人特別研究員
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| Project Period (FY) |
2017-11-10 – 2019-03-31
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| Keywords | channel surfaces / Guichard nets |
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| Outline of Annual Research Achievements |
Results were obtained in two areas, channel surfaces and Guichard nets.
The paper 'Channel surfaces in Lie sphere geometry' [1] with Pember will appear in Beitraege zur Algebra und Geometrie.
Discrete channel surface theory was completed. We characterized isothermic discrete channel surfaces analogous to the smooth case. The paper 'Discrete channel surfaces' [2] together with Hertrich-Jeromin and Rossman is near completion.
The geometry of smooth Guichard nets was further investigated. In particular, Egorov-Guichard nets and Guichard nets with one family of totally umbilic coordinate surfaces is now understood.
Diagonal surfaces of a triply orthogonal system were studied. An analytic condition for Guichard nets with minimal diagonal surfaces was proven, with explicit examples constructed.
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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
Using the obtained discretized channel surfaces, discrete cyclic triply orthogonal systems are definable. Aiming at useful geometric definitions of Guichard nets, we study discrete cyclic Guichard nets, a well-known class in the smooth setting.
To obtain ideas for defining discrete Guichard nets, we need explicit examples of smooth Guichard nets having surface classes with well-defined discrete counterparts. As there exists a rich theory for discrete minimal surfaces, Guichard nets with diagonal minimal surfaces are helpful and promising.
Another strategy is to discretize the associated systems of Guichard nets. For a relatively simple geometry, we start with Guichard nets with one family of totally umbilic coordinate surfaces. We use obtained geometric insights from the smooth case.
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| Strategy for Future Research Activity |
Using techniques from discrete geometry, we will discretize subclasses of Guichard nets to get ideas for a general discrete definition:
- the discrete Weierstrass representation will be used to obtain discrete Guichard nets with minimal surfaces as diagonal surfaces
- parallel families of discrete linear Weingarten surfaces will be used to construct discrete cyclic Guichard nets
As a starting point for a new collaboration, Thilo Roerig (TU Berlin) will visit in June. We plan to gain better geometric insights into the discrete transformation theory of channel surfaces. Hence, for this work we will rely on [1] and [2] above.
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