Geometry of surfaces in space forms

空间形式的表面几何

• 批准号: 11640080
• 负责人: UMEHARA Masaaki
• 金额: $ 2.3万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640080/
• 关键词: Mean Curvature; Surface; hyperbolic space; Total curvature

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.

我们得到以下结果：1。头部研究员Umehara对蜂毛的3个空间H^3的完全恒定平均曲率1表面（即CMC-1表面）进行了分类，该总绝对曲率（分别是双重绝对曲率）小于或等于4π。此外，当表面的总曲率小于或等于8π时，他给出了不存在和存在结果。这些结果在与Rossman和Yamada.2的联合工作中显示。首席调查员Umehara，Kokubu，Takahashi和Yamada给出了一个表面理论，该表面具有圆形高斯在紧凑型半密布谎言组中的映射，这是H^3中CMC-1表面的概括，并显示了Chern-Osserman在Mienimal表面的类似物中的类似物。此外，他们给出了几个这种表面的非平凡示例，并且这些表面的平均曲率都与环境空间的截面曲率成正比。3。首席研究员Umehara和Bobenko研究了H^3中恒定平均曲率的单轨道，并表明在给定的Riemannian 2-manifold上，具有规定的常量平均曲率在H^3中的等轴测浸入数量是最终的。

项目成果

A.Fujioka et al.: "Spacelike surfaces with harmonic inverse mean curvature"J.Math.Sci.Univ.Tokyo. 7. 657-698 (2000)

A.Fujioka 等人：“具有调和逆平均曲率的类空间表面”J.Math.Sci.Univ.Tokyo。

M.Umehara, et al.: "Metrics of constant curvature 1 with three conical singularities on the 2-sphere"Illinois Journal of Mathematics. 44. 72-94 (2000)

M.Umehara 等人：“2 球面上具有三个圆锥奇点的常曲率 1 的度量”《伊利诺斯数学杂志》。

H.Kanno, et al.: "Octonionic Yang-Mills instanton on quaternionic line bundle of Spin(7) holonomy"J.Geom.Phys.. 34. 302-320 (2000)

H.Kanno 等人：“Spin(7) 完整性四元线束上的八元杨-米尔斯瞬时子”J.Geom.Phys.. 34. 302-320 (2000)

R.Aiyama, et al.: "A global correspndence between CMC-surfaces in S and pairs of non-conformal harmonic maps into S"Proc.Amer.Math.Soc.. 128. 939-941 (2000)

R.Aiyama 等人：“S 中的 CMC 表面和成对的非共形谐波映射到 S 中的全局对应关系”Proc.Amer.Math.Soc.. 128. 939-941 (2000)

A.Fujioka et.al.: "Spacelike surface with harmonic inverse mean curvature"J.math.Sci.Univ.Tokyo. 7. 657-698 (2000)

A.Fujioka 等人：“具有调和逆平均曲率的类空间表面”J.math.Sci.Univ.Tokyo。