1998 Fiscal Year Final Research Report Summary
Representation formulas of Weierstrass type in submanifold theory
| Project/Area Number |
09640120
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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 |
Geometry
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| Research Institution | Kumamoto University |
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Principal Investigator |
INOUE Hisao Kumamoto Univ., Lect., 理学部, 講師 (40145272)
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| Co-Investigator(Kenkyū-buntansha) |
MAEBASHI Toshiyuki Kumamoto Univ., Prof., 理学部, 教授 (90032804)
OHWAKI Shin-ichi Kumamoto Univ., Prof., 理学部, 教授 (50040506)
KUROSE Takashi Fukuoka Univ., Assoc.Prof., 理学部, 助教授 (30215107)
HARAOKA Yoshishige Kumamoto Univ., Assoc.Prof., 理学部, 助教授 (30208665)
YAMADA Kotaro Kumamoto Univ., Assoc.Prof., 理学部, 助教授 (10221657)
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| Project Period (FY) |
1997 – 1998
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| Keywords | minimal surfaces / Weierstrass representation / monodoromoy / ODE / integrable system |
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| Research Abstract |
We investigated a global properties of Weierstrass representation. First of all, we studied a fundamental problem related to global problems. In particular, as the monodromy problem for minimal surfaces in Euclidean geometry is considered as a period problem of certain integral of holomorphic forms, that of CMC-1 surfaces in hyperbolic space can be considerd as a monodromy problem of a ordinary differential equation on Riemann surfaces. In this context, the monodromy problem is the condition for SL(2, C)-monodromy group to be reduced to the unitary group. To find a criterion of such a condition is difficult. However, when a problem satisfies some symmetric properties, it can be solved explicitly. Using this fact, we have constructed a large amount of examples of CMC-1 surfaces. Related to this problem, we investigated metrics of constant positive curvature with conical singularities, and obtained a classification result.
Related to classification of CMC-1 surfaces, we defined a homology invariant on CMC-1 surface, which is called flux, and proved some non-existence theorem using this invariant. Moreover, using Weierstrass representation for maximal surfaces in Minkowski 3-space, we have classified of surfaces with cone-like singularities with some finiteness.
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