Research for manifolds with conformal structure

共形结​​构流形的研究

基本信息

  • 批准号:
    09440044
  • 负责人:
  • 金额:
    $ 4.22万
  • 依托单位:
  • 依托单位国家:
    日本
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
  • 财政年份:
    1997
  • 资助国家:
    日本
  • 起止时间:
    1997 至 1999
  • 项目状态:
    已结题

项目摘要

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 … More 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
1.共形平坦超曲面。我们研究了四维空间形式中的共形平坦超曲面,并在四维标准球面上找到了每个超曲面的良好结构。根据共形平坦超曲面的结构,将共形平坦超曲面集分为三类:抛物类、椭圆类和双曲类。我们证明了这些类在球面的共形变换下是不变的,并且相应的类由在三维空间形式之一:欧氏空间、双曲空间或球面中由常曲率曲面构造的共形平坦超曲面组成.统计流形的共形-射影变换。本文得到了以下结果:统计流形的共形-射影变换保留了所有脐点和超曲面的Ricci曲率的反对称分量;而且,这个性质刻画了统计流形的共形-射影变换的性质,即保形-射影变换的性质 ...更多信息 当统计流形的维数大于2时的非形式投影变换。我们还发现了一个张量场,它在任何共形投影变换下都是不变的,并且如果基础统计流形是一个通常的黎曼流形,则它可以简化为共形曲率张量。三维空间形式中常平均曲率曲面(CMC曲面)的表示公式及其高斯映射。研究了目标空间为非正曲率下无界的非完备黎曼空间的调和映射的存在性问题。在这种情况下,我们证明了无穷远处Dirichlet问题调和映射的存在唯一性定理。作为应用,我们构造了三维双曲空间形式的CMC曲面.从可积系统理论的观点推广CMC曲面类。本文在三维空间形式中定义了具有调和逆平均曲率的曲面(HIMC曲面),并证明了这类曲面之间存在类似于Lawson对应的对应关系,这是CMC曲面类的特征之一.从变分的角度研究了HIMC曲面类与CMC曲面类的推广--H-曲面类之间的关系。证明了在一定的约化条件下,HIMC曲面是由H曲面的规范理论方程得到的。少

项目成果

期刊论文数量(0)
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K.Yamada: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space,and applications(with W.Rossman and M.Umehara)"Proc.Amer.Math.Soc.. 127. 2147-2154 (1999)
K.Yamada:“双曲 3 空间中平均曲率 1 表面的新通量及其应用(与 W.Rossman 和 M.Umehara)”Proc.Amer.Math.Soc.. 127. 2147-2154 (1999)
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K. Akutagawa: "A global correspondence between CMC-surfaces in SィイD13ィエD1 and pairs non-conformal harmonic maps into SィイD12ィエD1"Proc. Amer. Math. Soc.. 128. 939-941 (2000)
K. Akutakawa:“SIID13D1 中的 CMC 表面与 SIID12D1 中的非共形调和映射之间的全局对应关系”Proc.Amer. 128. 939-941 (2000)
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K.Shiohama: "Conformally flat 3-manifolds with constant scalar curvature"J.Math.Soc.Japan. 51. 209-226 (1999)
K.Shiohama:“具有恒定标量曲率的共形平坦 3 流形”J.Math.Soc.Japan。
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4.山田 光太郎: "Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus,(joint work with W.Rossman & M.Umehara)" Tohoku Math.J.49. 449-484 (1997)
4. Kotaro Yamada:“具有正亏格的双曲空间中的不可约常数平均曲率 1 曲面,(与 W.Rossman 和 M.Umehara 合作)”Tohoku Math.J.49(1997)。
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K. Akutagawa: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2 (with R. Aiyama, R. Miyaoka and M. Umehara)"Proc. Amer. Math. Soc.. 128. 939-941 (2000)
K. Akutakawa:“S^3 中的 CMC 表面与成对的非共形谐波映射到 S^2 之间的全局对应关系(与 R. Aiyama、R. Miyaoka 和 M. Umehara)”Proc。
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SUYAMA Yoshihiko其他文献

SUYAMA Yoshihiko的其他文献

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{{ truncateString('SUYAMA Yoshihiko', 18)}}的其他基金

Research on conformally flat hypersurfaces
共形平坦超曲面研究
  • 批准号:
    21540102
  • 财政年份:
    2009
  • 资助金额:
    $ 4.22万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Geometric structures on submanifolds in space forms and differential equations related to their structures.
空间形式子流形上的几何结构以及与其结构相关的微分方程。
  • 批准号:
    16340020
  • 财政年份:
    2004
  • 资助金额:
    $ 4.22万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
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