2000 Fiscal Year Final Research Report Summary
Construction of submanifold with constant mean curvature, and its applications
| 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)
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| Project Period (FY) |
1998 – 2000
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| Keywords | minimal surfaces / Weierstrass representation / CMC-1 surface / Gauss map / Osserman inequality / Total curvature |
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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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