2006 Fiscal Year Final Research Report Summary
The global behavior of curves and surfaces in space forms
| Project/Area Number |
15340024
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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 | Osaka University |
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Principal Investigator |
UMEHARA Masaaki Osaka Univ., Graduate School of Science, Professor, 理学研究科, 教授 (90193945)
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| Co-Investigator(Kenkyū-buntansha) |
KOISO Norihito Osaka Univ., Graduate School of Science, Professor, 理学研究科, 教授 (70116028)
YAMADA Kotaro Kyushu Univ., Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (10221657)
ROSSMAN Wayne F Kobe Univ., Faculty of Science, Associate Professor, 理学部, 助教授 (50284485)
KOKUBU Masatoshi Tokyo Denki Univ., School of Engineering, Associate Professor, 工学部, 助教授 (50287439)
INOGUCHI Junichi Utsnomiya Univ., Department of Math.Education, Associate Professor, 教育学部, 助教授 (40309886)
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| Project Period (FY) |
2003 – 2006
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| Keywords | Gaussian curvature / Singular point / Inflection point / Wave front |
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| Research Abstract |
We get the following results :
1.A maximal surface which is given by the real part of holomorphic isotropic immersion into C^3 is called a maxface. As a joint work with K.Yamada, the head investigator Umehara gave a Weierstrass-type representation formula for maxfaces, and gave an Osserman-type ineqality for complete maxfaces. The equality holds if and only if all ends of the surfaces are properly embedded. Moreover, as a joint work with K.Saji, S.Fujimori, and K.Yamada, the head investigator Umehara gave a criterion for the cuspidal cross cap, and showed that generic singular points for maxfaces consists of cuspidal edge, swallowtail and cuspidal cross cap.
2.As a joint work with K.Saji and K.Yamada, the head investigator Umehara studied the behavior of Gaussian curvature near the cuspidal edge and the swallowtail. In particular, the new geometric invariant on cuspidal edges called the singular curvature is introduced, and show that the integration of the singular curvature on the singular set is closely related to the Euler number of the surface.
3.A curve γ in the real projective plane is called anti-convex if for each point p on the curve, there exists a line passing through the point which does not meet y other than p. As a joint work with G.Thorbergsson, the head investigator Umehara studied the inflection points on anti-convex curves, and showed that the number of inflection points I and the number of the independent double tangents D satisfies the relation I-2D=3.
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