Research on constant mean curvature surfaces

常平均曲率曲面的研究

• 批准号: 12440012
• 负责人: KENMOTSU Katsuei
• 金额: $ 9.79万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440012/
• 关键词: minimal surface; constant mean curvature surface; mean curvature vector; complex space form; periodic surface of revolution; Delaunay surface; Bezier curve; harmonic map; 平均曲率一定; C-多様体; エネルギー最小写像; 周期的平均曲率; 周期的回転超曲面; ウィルモア汎関数; 多重Kaehler角度; P

1.We classified these isometric minimal immersions of real two dimensional Riemannian manifold with constant Gaussian curvature into a complex two dimensional complex space form. In this research, it was important to prove the constancy of the Kaehler angles of these immersions.2.We studied periodic surfaces of revolution in the three dimensional Euclidean space in order to understand the importance of constant mean curvature surfaces and got the criterion for a periodic function to be the mean curvature of a periodic surface of revolution. Moreover, we found an interesting relation between periodic mean curvature function and Bezier curves.3.Since the theory of non zero constant mean curvature surfaces is well developed on these twenty years, I have written a textbook about this subject in order to make a unified explanation of the theory. Recently, this book was translated into English by American Mathematical Society.

1.我们将这些等距的最小浸入真正的二维riemannian歧管的最小浸入中，恒定的高斯曲率将其分为复杂的二维复合空间形式。在这项研究中，重要的是要证明这些沉浸液的Kaehler角度的恒定。2。我们研究了三维欧几里得空间中革命的周期性表面，以了解恒定平均曲率表面的重要性，并获得了定期功能的标准，以使周期性的曲率是革命的平均曲率。此外，我们发现了周期性平均曲率函数与曲线更有趣的关系。3。由于非零常数平均曲率表面的理论在这20年中得到了很好的发展，我已经写了一本有关该主题的教科书，以便对理论进行统一的解释。最近，这本书被美国数学学会翻译成英语。

项目成果

Katsuei Kenmotsu: "Surfaces with constant mean curvature, Translations of Math.Monographs."Amer.Math.Soc.. 142 (2003)

Katsuei Kenmotsu：“具有恒定平均曲率的表面，Math.Monographs 的翻译。”Amer.Math.Soc.. 142 (2003)

Y.Ohnita: "Hamiltonian stability of certain minimal Lagrangian submanifolds"Tohoku Math.J.. 55. 583-610 (2003)

Y.Ohnita：“某些最小拉格朗日子流形的哈密顿稳定性”Tohoku Math.J.. 55. 583-610 (2003)

Katsuei Kenmotsu: "Energy minimizer maps on C-manifolds"Diff.Geom.Appl.. (印刷中). (2004)

Katsuei Kenmotsu：“C 流形上的能量最小化图”Diff.Geom.Appl..（印刷中）。

S.Nishikawa: "Harmonic maps in complex Finsler geometry, Geometric Variational Problems"Scans and Geometric Flows, Birkhauser. (in press). (2004)

S.Nishikawa：“复杂芬斯勒几何中的调和图，几何变分问题”扫描和几何流，Birkhauser。

砂田利一: "分割の幾何学"日本評論社. 140 (2000)

砂田理一：《除法几何》日本孝论社 140 (2000)。