| 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)
長瀬 道弘 大阪大学, 大学院・理学研究科, 教授 (70034733)
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| Project Period (FY) |
2002 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥31,460,000 (Direct Cost: ¥24,200,000、Indirect Cost: ¥7,260,000)
Fiscal Year 2005: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2004: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2003: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000)
Fiscal Year 2002: ¥8,060,000 (Direct Cost: ¥6,200,000、Indirect Cost: ¥1,860,000)
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| Keywords | Variational problem / Evolution equation / curve / surface / 弾性曲線 / 極小曲面 / 非連結 / 境界 / 部分多様体 / 一般次元 / 極小部分多様体 / 平均曲率一定超曲面 / 運動方程式 / 解の滑らかさ |
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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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