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．首席研究员 Masaaki Umehara 证明了欧几里德 n 空间中最小曲面的 Osserman 不等式的等式成立当且仅当每个端点没有自交并且渐近于悬链线或平面时，这是与 M 的联合工作。 Kokubu 和 K. Yamada。此外，我们构造了一个新的例子，它达到了等式。 2．首席研究员 Masaaki Umehara 与 W. Rossman 和 K. Yamada 合作，在双曲 3 空间中给出了可数个新的不可约常平均曲率一（即 CMC-1）曲面，其两端均不规则。3。首席研究员 Masaaki Umehara 初步证明了双曲 3 空间中 CMC-1 曲面的 Small 表示公式，并得到了类似的双曲 3 空间中平面表示公式，这是 M. Kokubu 和 K 的共同工作山田。双曲 3 空间中的平面如果承认奇点但可以提升到双曲 3 空间的单位余切丛中的勒让德浸入，则称为平面前沿。我们展示了具有完整末端的平坦前沿满足关于双曲高斯图的次数的奥瑟曼型不等式。当且仅当所有端点都没有自交时，等式才成立。此外，我们对具有嵌入规则 3 端的零属的平坦前端进行分类，并且还构建了具有五个规则嵌入端的属一的示例。

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。