1998 Fiscal Year Final Research Report Summary
A Modern Approach to Geometry of Curves and Surtaces and its Applications
| Project/Area Number |
09640106
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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 | HIROSHIMA UNIVERSITY (1998)
Osaka University (1997) |
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Principal Investigator |
UMEHARA Masaaki Fac.Sci., HIROSHIMA UNIVERSITY Prof., 理学部, 教授 (90193945)
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| Co-Investigator(Kenkyū-buntansha) |
KOWATA Asutaka Fac.Sci., HIROSHIMA UNIVERSITY Research Associate, 理学部, 助手 (50033931)
DOI Hideo Fac.Sci., HIROSHIMA UNIVERSITY Lecturer, 理学部, 講師 (50197993)
KOISO Norihito Osaka U., Grad.School Sci., Prof., 大学院・理学研究科, 教授 (70116028)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Cuvue / Suvface / Vertex / Constant mean cuvuature / Conlcal Singularity |
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| Research Abstract |
We get the following results :
1. The head investigator Masaaki Umehara gave a flux formula for surfaces of constant mean curvature 1 (i.e. CMC-1 surfaces) in the hyperbolic 3-space as an analogy for that of minimal surfaces in the Euclidean space, which is a joint work with Wayne Rossman (Kobe Univ.) and Kotaro Yamada (Kumamoto Univ.). As an application, he and K.Yamada classified all conformal metrics of constant curvature 1 with three conical singularities on 2-sphere.
2. The head investigator Masaaki Umehara and Gudlaugur Thorbergs-son (Koln Univ.) gave a refinement of the classical four vertex theorem on simple closed curves, where they gave an abstract intrinsic approach regarding the theorem as a kind of homology theory on S^1 whose first Betti number is the number of vertices. As an application, they prove a four vertex theorem on some kind of space curves and also gave a new proof of the four vertex theorem for diffeo-morphisms on S^1. Moreover, they generalized the theory for much higher order geometry and gave sharp estimates for affine vertices on simple closed curves.
3. The investigator Koiso investigated the elasticas in a Riemannian manifolds by analytic approach.
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