The global behavior of curves and surfaces in space forms

空间形式中曲线和曲面的全局行为

• 批准号: 15340024
• 负责人: UMEHARA Masaaki
• 金额: $ 6.66万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340024/
• 关键词: Gaussian curvature; Singular point; Inflection point; Wave front; 双曲型空間; ガウス写像; 平坦; 曲面; 曲線

We get the following results :1.A maximal surface which is given by the real part of holomorphic isotropic immersion into C^3 is called a maxface. As a joint work with K.Yamada, the head investigator Umehara gave a Weierstrass-type representation formula for maxfaces, and gave an Osserman-type ineqality for complete maxfaces. The equality holds if and only if all ends of the surfaces are properly embedded. Moreover, as a joint work with K.Saji, S.Fujimori, and K.Yamada, the head investigator Umehara gave a criterion for the cuspidal cross cap, and showed that generic singular points for maxfaces consists of cuspidal edge, swallowtail and cuspidal cross cap.2.As a joint work with K.Saji and K.Yamada, the head investigator Umehara studied the behavior of Gaussian curvature near the cuspidal edge and the swallowtail. In particular, the new geometric invariant on cuspidal edges called the singular curvature is introduced, and show that the integration of the singular curvature on the singular set is closely related to the Euler number of the surface.3.A curve γ in the real projective plane is called anti-convex if for each point p on the curve, there exists a line passing through the point which does not meet y other than p. As a joint work with G.Thorbergsson, the head investigator Umehara studied the inflection points on anti-convex curves, and showed that the number of inflection points I and the number of the independent double tangents D satisfies the relation I-2D=3.

我们得到以下结果：1.a最大表面由全体形态各向同性浸入c^3的实际部分称为maxface。作为与K.Yamada的联合合作，首席研究员Umehara为Maxfaces提供了Weierstrass型代表公式，并给出了Osserman型的INEQALITY，以实现完整的Maxfaces。当且仅当正确嵌入表面的所有末端时，平等才能保持。此外，作为与K.Saji，S.Fujimori和K.Yamada的联合合作，首席调查员Umehara给出了Cuspidal Cross上限的标准，并显示最大值的通用奇异点由Cuspidal Edgets，SwallowTaille和Cuspidal Cross Cap。研究了高斯曲率在尖边和燕尾附近的行为。 In particular, the new geometric invariant on cuspidal edges called the singular Curvature is introduced, and show that the integration of the singular curvature on the singular set is closely related to the Euler number of the surface.3.A curve γ in the real projective plane is called anti-convex if for each point p on the curve, there exists a line passing through the point which does not meet y other than p.作为与G.Thorbergsson的联合工作，负责人乌梅哈拉（Umehara）研究了对抗凸曲线的影响点，并表明影响点I的数量和独立双重切线D的数量满足了I-2D = 3的关系。

项目成果

Flat fronts in hyperbolic 3-space and their caustics

• DOI: 10.2969/jmsj/1180135510
• 发表时间: 2005-11
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: M. Kokubu;W. Rossman;M. Umehara;Kotaro Yamada

Constructing mean curvature 1 surfaces in H^3 with irregular ends.

在 H^3 中构造具有不规则端部的平均曲率 1 曲面。

• 发表时间: 2005
• 期刊: Global Theory of Minimal Surfaces (ed. D.Hoffman) Clay Mathematics Proceedings (A.M.S.) 2
• 作者: W.Rossmun;M.Umehara;K.Yamada
• 通讯作者: K.Yamada

M.Kokubu, et al.: "An elementrary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)"Osaka J.Math.. 40. 697-715 (2003)

M.Kokubu 等人：“PSL(2,C) 中零曲线的 Small 公式的基本证明和 PSL(2,C) 中 Legendrian 曲线的类似物”Osaka J.Math.. 40. 697-715

Maximal surfaces with singularities in Mikowski space

米科夫斯基空间中具有奇点的最大曲面

• 发表时间: 2006
• 期刊: Hokkaido Math.J. 35
• 作者: M.Umehara;K.Yamada
• 通讯作者: K.Yamada

Flat fronts in hyperholic 3-space

超霍尔 3 空间中的平坦前沿

• 发表时间: 2004
• 期刊: Pacific Journal of Mathematics 216
• 作者: M.Kokubu;et al.
• 通讯作者: et al.