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 为三维欧氏空间中的完备曲面。在所有边界为 P 且给定体积的浸没表面中，面积泛函的每个临界点称为支撑表面 P 的静止浸没。在 P 为平面的特殊情况下，我们证明任何稳定的静止浸没是嵌入到半球上。2。我们研究了结节的性质和形状，它们是具有自交的 Delaunay 曲面（三维欧几里得空间中具有恒定平均曲率的旋转曲面）3。我们获得了在三维欧几里德空间中具有恒定平均曲率（CMC）的浸没表面的充分条件，在该条件下，它具有固定边界的 CMC 变形。此外，我们还获得了 CMC 浸泡稳定性的标准。这两者都是通过使用与面积泛函的第二变体相关的特征值问题的特征值和特征函数的性质来实现的。在某种特殊情况下，结合这些结果，我们得到了一种“看得见的”判断稳定性的方法。 4．我们证明了众所周知的规则最小曲面面积函数的二阶变分公式对于“良好”变化的广义最小曲面（具有分支点的最小曲面）也有效。5．我们推导出了三维空间形式中具有恒定平均曲率的浸没表面强稳定的充分条件。