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.由全纯各向同性浸入C^3的实部给出的最大表面称为maxface 作为与K.Yamada的联合工作，首席研究员Umehara给出了Weierstrass型表示。 maxfaces 的公式，并给出了完整 maxfaces 的 Osserman 型不等式，当且仅当曲面的所有端部都正确嵌入时，该等式才成立。首席研究员 Umehara K.Saji、S.Fujimori 和 K.Yamada 给出了尖头十字帽的标准，并表明 maxfaces 的通用奇异点由尖头边缘、燕尾和尖头十字帽组成。 2. 作为关节首席研究员 Umehara 与 K.Saji 和 K.Yamada 合作，研究了尖缘和燕尾附近的高斯曲率行为。引入了新的尖边几何不变量——奇异曲率，表明奇异曲率在奇异集上的积分与曲面的欧拉数密切相关。3.实射影平面上的曲线γ称为反凸，如果对于曲线上的每个点 p，都存在一条穿过除 p 之外的不满足 y 的点的线 作为与 G.Thorbergsson 的联合工作，首席研究员 Umehara 研究了该点。反凸曲线上的拐点，并且表明拐点I的数量和独立双切线D的数量满足关系I-2D=3。

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.