| Project/Area Number |
10440024
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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 | Kyushu University (2000)
Kumamoto University (1998-1999) |
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Principal Investigator |
YAMADA Kotaro Kyushu University, Faculty of Math., Prof., 大学院・数理学研究院, 教授 (10221657)
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| Co-Investigator(Kenkyū-buntansha) |
ROSSMAN Wayne Kobe Univ., Fac.of Sci., Assoc.Prof., 理学部, 助教授 (50284485)
CHO Koji Kyushu University, Faculty of Math., Assoc.Prof., 大学院・数理学研究院, 助教授 (10197634)
YAMAGUCHI Takao Kyushu University, Faculty of Math., Prof., 大学院・数理学研究院, 教授 (00182444)
INOUE Hisao Kumamoto Univ., Fac.of Sci., Lect., 理学部, 講師 (40145272)
KUROSE Takashi Fukuoka Univ., Fac.of Sci., Assoc.Prof., 理学部, 助教授 (30215107)
大脇 信一 熊本大学, 理学部, 教授 (50040506)
伊藤 仁一 熊本大学, 教育学部, 助教授 (20193493)
前橋 敏之 熊本大学, 理学部, 教授 (90032804)
原岡 喜重 熊本大学, 理学部, 助教授 (30208665)
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| Project Period (FY) |
1998 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥5,900,000 (Direct Cost: ¥5,900,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | minimal surfaces / Weierstrass representation / CMC-1 surface / Gauss map / Osserman inequality / Total curvature / 非コンパクト型対称空間 / キーワード7 / キーワード8 / 低平均曲率曲面 / フラックス / モノドロミー問題 / 定平均曲率曲面 / 常微分方程式 / 可積分系 |
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| Research Abstract |
We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space.
As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π.
On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature.
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