Weierstrass-type representation formulas and their application to surfaces with singularities

Weierstrass型表示公式及其在奇点曲面上的应用

• 批准号: 21340016
• 负责人: YAMADA Kotaro
• 金额: $ 9.82万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009-04-01 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21340016/
• 关键词: 曲面; 特異点; フロント; 特異曲率; 連接接束; ワイエルストラス表現; 交叉帽子; カスプ辺; 双曲空間の平坦波面; 極大曲面; CMC-1 曲面; ガウス・ボンネの定理; 双曲計量; 共形平坦計量; 3ノイド; 超幾何微分方程式; wave fronts; coherent tangent bundles; front bundles; Gauss-Bonnet formula; duality; conformally flat; comnplet /; 曲面; 特異点; フロント; 特異曲率; 連接接束; ワイエルストラス表現; 交叉帽子; カスプ辺; 双曲空間の平坦波面; 極大曲面; CMC-1 曲面; ガウス・ボンネの定理; 双曲計量; 共形平坦計量; 3ノイド; 超幾何微分方程式; wave fronts; coherent tangent bundles; front bundles; Gauss-Bonnet formula; duality; conformally flat; comnplet /

We investigated, Weierstrass-type representation formula, global properties of several classes of surfaces with singularities, such as flat surfaces in hyperbolic 3-space, maximal surfaces in Minkowski 3-space, CMC-1 surfaces in de Sitter 3-space, and improper affine sphere in affine 3-space, and obtained a charctreization of completeness, Osserman-type inequalis etc.In addition, flat trinoids in hyperbolic space and CMC-1 2-noids in de Sitter 3-space are classified. On the other hand, as a basic tool of differential geometry of wave front, we introduced a notion of "sinular curvature" and investigated a rdelationship between singular curvature and behavior of Gaussian curvature. As a result, we obtained Gauss-Bonnet type formula for wave fronts. Moreover, as an intrinsic formulation of wave fronts, we introduced a notion of "coherent tangent bundles" and gave an application of their duality.

We investigated, Weierstrass-type representation formula, global properties of several classes of surfaces with singularities, such as flat surfaces in hyperbolic 3-space, maximal surfaces in Minkowski 3-space, CMC-1 surfaces in de Sitter 3-space, and improper affine sphere in affine 3-space, and obtained a charctreization of completeness, Osserman-type inequalis等等。此外，对螺母空间中的平坦三明治定位和de Sitter 3空间中的CMC-1 2个子素进行了分类。另一方面，作为波浪前差几何形状的基本工具，我们引入了“曲曲率”的概念，并研究了高斯曲率的奇异曲率和行为之间的划分。结果，我们获得了波浪前的高斯 - 邦网型公式。此外，作为波浪锋的内在表述，我们引入了“连贯的切线束”的概念，并给出了它们的双重性。