2005 Fiscal Year Final Research Report Summary
Variational problem and evolution equation of curves and surfaces
| Project/Area Number |
14204004
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | Osaka University |
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Principal Investigator |
KOISO Norihito Osaka university, Graduate school of sciences, Professor, 大学院・理学研究科, 教授 (70116028)
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| Co-Investigator(Kenkyū-buntansha) |
MABUCHI Toshiki Osaka university, Graduate school of sciences, Professor, 大学院・理学研究科, 教授 (80116102)
NISHITANI Tatsuo Osaka university, Graduate school of sciences, Professor, 大学院・理学研究科, 教授 (80127117)
UMEHARA Masaaki Osaka university, Graduate school of sciences, Professor, 大学院・理学研究科, 教授 (90193945)
ENOKI Ichiro Osaka university, Graduate school of sciences, Associate Professor, 大学院・理学研究科, 助教授 (20146806)
GOTO Ryuji Osaka university, Graduate school of sciences, Associate Professor, 大学院・理学研究科, 助教授 (30252571)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Variational problem / Evolution equation / curve / surface |
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| Research Abstract |
The most natural variational problem of closed submanifolds in the 3-euclidean space is the elastic curves in 1-dimensional case, and the constant mean curvature surfaces in 2-dimensional case. These problems are extended naturally to n-euclidean spaces as curves or hyper surfaces. However, we don't have good variational problem of closed mid-dimensional submanifold in n-dimensional euclidean spaces In this research, we defined the following new good variational problem. Consider pairs (S, dS) of minimal submanifold S and its boundary dS. Given the volume of dS, we seek a minimal submanifold S whose volume attains maximum. A solution of this variational problem is called a max-min submanifold. We got the following results.
1. The pair of a totally geodesic submanifold and its constant mean curvature surfaces is max-min submanifold.
2. In particular, a round sphere of any dimensional Euclidean space with any codimension is max-min submanifold.
3. The solution (2) is stable.
4. A pair of a minimal cone C and the intersection of C and the unit sphere is max-min submanifold.
5. We can construct non-homogeneous examples in the torus.
6. Since the variational problem is conditional, two stabilities are defined. Let k be the trace of second fundamental form for outer unit vector. If k is positive, then the solution is A-unstable. If k is negative, then A-stability and B-stability are equivalent.
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