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Research on constant mean curvature surfaces

常平均曲率曲面的研究

基本信息

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

项目摘要

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.我们将这些具有恒定高斯曲率的实二维黎曼流形的等距最小浸入分类为复二维复空间形式。在本研究中，证明这些浸没的凯勒角的恒定性非常重要。2.我们研究了三维欧几里得空间中的周期旋转曲面，以了解恒定平均曲率表面的重要性，并得到了周期函数作为周期旋转表面的平均曲率的判据。此外，我们还发现了周期平均曲率函数与贝塞尔曲线之间的一个有趣的关系。3.由于非零常平均曲率曲面理论在这二十年来已经得到了很好的发展，我写了一本关于这个主题的教科书，以便对该理论做出统一的解释。近日，本书被美国数学会翻译成英文。