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DEFORMATION AND STABILITY OF SURFACES WITH CONSTANT MEAN CURVATURE

平均曲率恒定的表面的变形和稳定性

基本信息

批准号：
11640200
负责人：
KOISO Miyuki
金额：
$ 1.47万
依托单位：
KYOTO UNIVERSITY OF EDUCATION
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640200/
关键词：
CONSTANT MEAN CURVATURE SURFACE MINIMAL SURFACE STABILITY OF CONSTANT MEAN CURVATURE SURFACE STABILITY OF MINIMAL SURFACE SECOND VARIATION FORMULA FREE BOUNDARY PROBLEM DELAUNAY SURFACE 第2変分公式 Weierstrass公式ガラス写像

项目摘要

1. Let P be a complete surface in the threee-dimensional euclidean space. Each critical point of the area functional, among all immersed surfaces with boundary in P and with a given volume, is called a stationary immersion for supporting surface P.In the special case where P is a plane, we proved that any stable stationary immersion is an embedding onto a hemisphere.2. We studied properties and shapes of nodoids, which are the surfaces of Delaunay (surfaces of revolution with constant mean curvature in the threee-dimensional euclidean space) with self-intersections3. We obtained sufficient conditions for a immersed surface with constant mean curvature (CMC) in the threee-dimensional euclidean space under which it has a CMC-deformation that fixes the boundary. Moreover, we obtained a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of the eigenvalues and the eigenfunctions of the eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtained a 'visible' way of judging the stability.4. We proved that the well-known second variation formula of the area function for regular minimal surfaces is valid also for generalized minimal surfaces (minimal surfaces with branch points) for 'good' variations.5. We derive sufficient conditions for immersed surfaces with constant mean curvature in three-dimensional space forms to be strongly stable.

1.设P是三维欧氏空间中的完备曲面。在所有边界在P中且具有给定体积的浸入曲面中，面积泛函的每个临界点称为支撑曲面P的稳定浸入.在P是平面的特殊情况下，我们证明了任何稳定的稳定浸入都是到半球上的嵌入.我们研究了具有自相交的Delaunay曲面（三维欧氏空间中具有常平均曲率的旋转曲面）的节点面的性质和形状。本文给出了三维欧氏空间中具有常平均曲率（CMC）的浸入曲面具有固定边界的CMC-变形的充分条件。此外，我们还得到了CMC浸泡稳定性的判据。这两个都是通过使用与面积泛函的第二变分相关联的本征值问题的本征值和本征函数的性质来实现的。在某种特殊情况下，将这些结果结合起来，得到了一种直观的稳定性判断方法.我们证明了著名的正则极小曲面面积函数的二次变分公式对于广义极小曲面（具有分支点的极小曲面）的“好”变分也有效。5.给出了三维空间形式中具有常平均曲率的浸入曲面强稳定的充分条件。