Research on surfaces of constant mean curvature one in hyperbolic space and its application

双曲空间中常平均曲率曲面的研究及其应用

• 批准号: 13640075
• 负责人: UMEHARA Masaaki
• 金额: $ 2.56万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640075/
• 关键词: Hyperbolic space; Mean curvature; Gauss map; Gaussian curvature

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.

我们得到以下结果：1。首席调查员Masaaki Umehara证明，当且仅当每个端都没有自我交流和渐近性的catenoid或平面渐近时，Osserman不平等的平等才能保持最小表面，这是与M. Kokubu和K. Yamada的联合工作。此外，我们构建了一个达到平等的新示例2。负责人Masaaki Umehara在双曲线3空间中提供了许多新的不可还原常数平均曲率（即CMC-1）表面，其末端全部不规则，这是与W. Rossman和K. Yamada.3的共同作品。首席研究员Masaaki Umehara给出了柔曲3空间中CMC-1表面的小表面的基本证明，并且在双曲线3空间中的平面表面有类似的表示公式，这是M. Kokubu和K. Yamada的联合工作。如果柔双波利三空间的平坦表面被称为平坦的锋，则可以将其抬高，但可以将其浸入legendrian浸入单元中的单位三个空间捆绑包中。我们展示了完整末端的平坦前部满足了双曲线高斯图的程度的Osserman型不等式。当且仅当所有目的都没有自我交集时，平等才能保持。此外，我们将零属的平坦前端与嵌入的常规3端进行了分类，还构建了一个属的示例，其中五个常规嵌入端。

项目成果

A. I. Bobenko and M. Umehara: "Monodromy of isometric deformation of CMC-surfaces"Hiroshima Math. J.. 31. 291-297 (2001)

A. I. Bobenko 和 M. Umehara：“CMC 表面等距变形的单律”广岛数学。

M.Kokubu, M.Takahashi, M.Umehara K.Yamada: "An analogue of minimal surface theory in SL(n, C)/Su(n)"Trans. Amer. Math. Soc.. 354. 1299-1325 (2002)

M.Kokubu、M.Takahashi、M.Umehara K.Yamada：“SL(n, C)/Su(n) 中最小曲面理论的类似物”Trans。

M.Kokubu, M.Takahashi, M.Umehara, K.Yamada: "An analogue of minimal surface theory in SL(n, C)/SU(n)"Trans. Amer. Math. Soc.. 354. 1299-1325 (2002)

M.Kokubu、M.Takahashi、M.Umehara、K.Yamada：“SL(n，C)/SU(n) 中最小曲面理论的类似物”Trans。

M. Kokubu, M. Takahashi, M. Umehara, and K. Yamada,: "An analogue of minimal surface theory in SL(n, C) / SU(n)"Trans. Amer. Math. Soc.,. 354. 1299-1325 (2002)

M. Kokubu、M. Takahashi、M. Umehara 和 K. Yamada，：“SL(n, C) / SU(n) 中最小曲面理论的类似物”Trans。

M. Kokubu, M. Umehara, and K. Yamada: "Minimal surfaces that attain equality in the Chern-Osserman inequality"Contemporary Mathematics. 303. 223-228 (2002)

M. Kokubu、M. Umehara 和 K. Yamada：“在 Chern-Osserman 不等式中获得相等的最小曲面”当代数学。