2000 Fiscal Year Final Research Report Summary
Geometry of surfaces in space forms
| Project/Area Number |
11640080
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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 |
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Principal Investigator |
UMEHARA Masaaki Hiroshima Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90193945)
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| Co-Investigator(Kenkyū-buntansha) |
HONDA Nobuhiro Hiroshima Univ., Graduate School of Science, Research Associate, 大学院・理学研究科, 助手 (60311809)
KANNO Hiroaki Hiroshima Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90211870)
MATSUMOTO Takao Hiroshima Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025467)
INOGUCHI Junichi Fukuoka Univ., Faculty of Science, Research Associate, 理学部, 助手 (40309886)
KOKUBU Masatoshi Tokyo Denki Univ., Department of Natural Science, Lecturer, 工学部, 講師 (50287439)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Mean Curvature / Surface / hyperbolic space / Total curvature |
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| Research Abstract |
We get the following results :
1. The head investigator Umehara gave a classification for complete constant mean curvature 1 surfaces (i.e. CMC-1 surfaces) in the hyperbolic 3-space H^3 of total absolute curvature (resp. the dual total absolute) curvature less than or equal to 4π. Moreover, he gave non-existence and existence results when the surfaces has dual total curvature less than or equal to 8π. These results are shown in a joint work with Rossman and Yamada.
2. The head investigator Umehara, Kokubu, Takahashi and Yamada gave a theory of surfaces with holomorphic Gauss maps in the duals of compact semisimple Lie groups, which is a generalization of CMC-1 surfaces in H^3, and show an analogue of Chern-Osserman Inequality for minimal surfaces in the Euclidean π-space. Moreover, they gave several non-trivial examples of such surfaces and showed mean curvature of these surfaces are all proportional to the sectional curvature of the ambient space.
3. The head investigator Umehara and Bobenko investigated the monodromy of constant mean curvatures in H^3 and showed that the number of isometric immersions with a prescribed constant mean curvature into H^3 on a given Riemannian 2-manifold is finite.
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