1999 Fiscal Year Final Research Report Summary
Research for manifolds with conformal structure
| Project/Area Number |
09440044
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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 | FUKUOKA UNIVERSITY |
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Principal Investigator |
SUYAMA Yoshihiko Fukuoka Univ., Fac. Sci., Prof., 理学部, 教授 (70028223)
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| Co-Investigator(Kenkyū-buntansha) |
KUROSE Takashi Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (30215107)
AKUTAGAWA Kazuo Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (80192920)
SHIOHAMA Katsuhiro Fukuoka Univ., Fac. Sci. & Engin., Prof., 理工学部, 教授 (20016059)
INOGUCHI Jun-ichi Fukuoka Univ., Fac. Sci., Assist. Prof., 理学部, 助手 (40309886)
YAMADA Kataro Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (10221657)
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| Project Period (FY) |
1997 – 1999
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| Keywords | conformal structure / conformally flat hypersurface / conformally flat manifold / statistical manfold / conformal-projective transformation / constant mean curvature surface / harmonic map / surface with harmonic inverse mean curvature |
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| Research Abstract |
1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere.
2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the co
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nformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold.
3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form.
4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction. Less
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