Development of analysis and discretization in differential geometry

微分几何分析和离散化的发展

• 批准号: 20K03585
• 负责人: Rossman W.F
• 金额: $ 2.75万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2020
• 资助国家: 日本
• 起止时间: 2020-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20K03585/
• 关键词: 離散的微分幾何学; 離散曲面; 離散曲線; 特異点; Darboux変換; differential geometry; surface theory; discretization; integrable systems; analysis

This is research on discrete curves and surfaces established by the use of transformation theory, a theory that can be viewed as a discrete construction within the smooth category that allows for natural notions of discretization. Previously understood discrete isothermic surfaces have been extended here to the wider class of discrete Omega surfaces, together with a transformation theory for this wider class that preserves underlying structures in the corresponding smooth case. In particular, Darboux transformations for the full class have been developed. Additionally, discretization of the potential mKdV equation has been seen from the perspective of Darboux transformations of curves. Also, in a related topic, singularities and signature changes of smooth surfaces in de Sitter 3-space have been explored, with particular application to various catenoids in that space.

这是对使用变换理论建立的离散曲线和曲面的研究，该理论可以被视为光滑类别中的离散结构，允许自然的离散化概念。以前理解的离散等温曲面在这里被扩展到更广泛的离散欧米茄曲面，以及在相应光滑情况下保留底层结构的这一更广泛类别的变换理论。特别是，已经开发了整个类的达布变换。另外，从曲线的达布变换的角度，对势能mKdV方程进行了离散化。此外，在一个相关的主题中，研究了de Sitter三维空间中光滑表面的奇异性和特征变化，并特别应用于该空间中的各种链状体。

项目成果

Applications of Harmonic Maps and Higgs Bundles in Differential Geometry, Kyoto

调和图和希格斯丛在微分几何中的应用，京都

• 发表时间: 2022

Broadening discrete surface classes

拓宽离散曲面类别

• 发表时间: 2022
• 作者: F. Burstall;U. Hertrich-Jeromin;W. Rossman and S. Santos;Shin-ichi Ohta;Wayne Rossman;Shin-ichi Ohta;Wayne Rossman
• 通讯作者: Wayne Rossman

Omega surfaces, with relations to singularities and discretization

欧米茄曲面，与奇点和离散化的关系

• 发表时间: 2022
• 作者: F. Burstall;U. Hertrich-Jeromin;W. Rossman and S. Santos;Shin-ichi Ohta;Wayne Rossman;Shin-ichi Ohta;Wayne Rossman;Wayne Rossman
• 通讯作者: Wayne Rossman

Korea University(韓国)

高丽大学（韩国）

Differential Geometry of Curves and Surfaces with Singularities

• DOI: 10.1142/12284
• 发表时间: 2021-02
• 作者: M. Umehara;K. Saji;Kotaro Yamada;W. Rossman