2006 Fiscal Year Final Research Report Summary
Application of integrable systems methods to surfaces with particular variational properties
| Project/Area Number |
15340023
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Kobe University |
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Principal Investigator |
ROSSMAM W.F. Kobe University, Faculty of Science, Assoc. Professor, 理学部, 助教授 (50284485)
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| Co-Investigator(Kenkyū-buntansha) |
OHNITA Yoshihiro Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (90183764)
GUEST M. Tokyo Metropolitan University, Graduate School of Science, Professor, 東京・大学院理学研究科, 教授 (10295470)
YAMADA Kotaro Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (10221657)
KOKUBU Masatoshi Tokyo Denki University, Department of Natural Science, Assoc., Professor, 工学部, 助教授 (50287439)
INOGUCHI Jun-ichi Utsunomiya University, Faculty of Education, Assoc., Professor, 教育学部, 助教授 (40309886)
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| Project Period (FY) |
2003 – 2006
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| Keywords | constant mean curvature / integrable systems / Euclidean space / 3-dimensional spherical space / hyperbolic space / 平坦曲面 |
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| Research Abstract |
The following results were obtained:
1) In a joint research project with U. Hertrich-Jeromin, S. Santos and F. Burstall, a suitable definition for discrete constant mean curvature surfaces in 3 dimensional space forms was obtained. Those 3 dimensional space forms consist of Euclidean 3-space, spherical 3-space and hyperbolic 3-space. It was shown that this new definition matches the old definition that is known for the Euclidean case, and this definition is new in the hyperbolic case. Using this definition, discrete Delaunay surfaces were studied, along with their discrete Darboux and Backlund transformations. An important tool in this research was the notion of conserved quantities. The case of smooth surfaces was developed by S. Santos and F. Burstall, while the discrete case was developed by U. Hertrich-Jeromin and myself.
2) In a joint research project with my Ph.D. graduate student N. Sultana, the stability and Morse index of constant mean curvature surfaces of revolution in spherical 3-space was studied. Because the axis of such a surface is a closed loop, these surfaces can become close tori, and then they will have finite index. It was shown that all such surfaces are unstable, and that they all have index at least 5, and (depending on the choice of surface) the index can be arbitrarily large. The index is the number of negative eigenvalues of the associated Jacobi operator.
3) In a continuation of a project with M. Kokubu, M. Umehara and K. Yamada, surfaces with constant Gauss curvature 0 in hyperbolic 3-space (flat fronts, which can have singularities) were studied. In particular, this year, it was shown that the caustics of such surfaces can have ends with asymptotic behavior described by cycloids.
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