2000 Fiscal Year Final Research Report Summary
DEFORMATION AND STABILITY OF SURFACES WITH CONSTANT MEAN CURVATURE
| Project/Area Number |
11640200
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | KYOTO UNIVERSITY OF EDUCATION |
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Principal Investigator |
KOISO Miyuki FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (10178189)
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| Co-Investigator(Kenkyū-buntansha) |
KOKUBU Masatoshi TOKYO DENKI UNIVERSITY, DEPARTMENT OF NATURAL SCIENCE, LECTURER, 工学部, 講師 (50287439)
AIYAMA Reiko (相山 玲子) UNIVERSITY OF TSUKUBA, INSTITUTE OF MATHEMATICS, LECTURER, 数学系, 講師 (20222466)
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| Project Period (FY) |
1999 – 2000
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| Keywords | CONSTANT MEAN CURVATURE SURFACE / MINIMAL SURFACE / STABILITY OF CONSTANT MEAN CURVATURE SURFACE / STABILITY OF MINIMAL SURFACE / SECOND VARIATION FORMULA / FREE BOUNDARY PROBLEM / DELAUNAY SURFACE |
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| Research Abstract |
1. Let P be a complete surface in the threee-dimensional euclidean space. Each critical point of the area functional, among all immersed surfaces with boundary in P and with a given volume, is called a stationary immersion for supporting surface P.In the special case where P is a plane, we proved that any stable stationary immersion is an embedding onto a hemisphere.
2. We studied properties and shapes of nodoids, which are the surfaces of Delaunay (surfaces of revolution with constant mean curvature in the threee-dimensional euclidean space) with self-intersections
3. We obtained sufficient conditions for a immersed surface with constant mean curvature (CMC) in the threee-dimensional euclidean space under which it has a CMC-deformation that fixes the boundary. Moreover, we obtained a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of the eigenvalues and the eigenfunctions of the eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtained a 'visible' way of judging the stability.
4. We proved that the well-known second variation formula of the area function for regular minimal surfaces is valid also for generalized minimal surfaces (minimal surfaces with branch points) for 'good' variations.
5. We derive sufficient conditions for immersed surfaces with constant mean curvature in three-dimensional space forms to be strongly stable.
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