Research on surfaces of constant mean curvature one in hyperbolic space and its application
| Project/Area Number |
13640075
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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) |
WAYNE Rossman Kobe Univ. Faculty of Science, Associate Professor, 理学部, 助教授 (50284485)
YAMADA Kotaro Hiroshima Univ. Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10221657)
MATSUMOTO Takao Hiroshima Univ. Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025467)
INOGUCHI Junichi Utsunomiya Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (40309886)
KOKUBU Masatoshi Tokyo Denki Univ., School of Engineering, Associate Professor, 工学部, 助教授 (50287439)
土井 英雄 広島大学, 大学院・理学研究科, 講師 (50197993)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | Hyperbolic space / Mean curvature / Gauss map / Gaussian curvature |
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| Research Abstract |
We get the following results:
1. The head investigator Masaaki Umehara proved that the equality of the Osserman inequality for minimal surfaces in Euclidean n-space holds if and only if each end has no self-intersection and asymptotic to a catenoid or a plane, which is a joint work with M. Kokubu and K. Yamada. Moreover, we construct a new example which attains the equality.
2. The head investigator Masaaki Umehara gave countable many new irreducible constant mean curvature one (i.e. CMC-1) surfaces in hyperbolic 3-space whose ends all irregular, which is a joint work with W. Rossman and K. Yamada.
3. The head investigator Masaaki Umehara gave an elementary proof of the Small's representation formula for CMC-1 surfaces in hyperbolic 3-space and also got a similar representation formula for flat surfaces in hyperbolic 3-space, which is a joint work M. Kokubu and K. Yamada. A flat surface in hyperbolic 3-space called a flat front if it admits singularity but can be lifted to a Legendrian immersion into the unit cotangent bundle of hyperbolic 3-space. We showed flat fronts with complete ends satisfy an Osserman-type inequality with respect to the degree of the hyperbolic Gauss maps. The equality holds if and only if all ends have no self intersection. Furthermore, we classify flat front of genus zero with embedded regular 3-ends and also construct an example of genus one with five regular embedded ends.
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Report
(3 results)
Research Products
(12 results)
