Functinal analytic approach to reseaches on deformation and stability of surfaces with constant mean curvature

常平均曲率曲面变形与稳定性研究的函数分析方法

• 批准号: 13640211
• 负责人: KOISO Miyuki
• 金额: $ 2.05万
• 依托单位: Nara Women's University (2003)Kyoto University of Education (2001-2002)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640211/
• 关键词: constant mean curvature; prescribed mean curvature; variational problem; Gauss map; stability; parametric elliptic functional; 調和逆平均曲率曲面; Lagrangian曲面; 変文問題; 平均曲率; Euler-Lagrange方程式; 平均局率一定曲面; prescribed mean curvature surface

1.We studied the variational theory of surfaces whose mean curvature is prescribed to be a linear function of their height above a horizontal plane (PMC surfaces). We developed a flux formula and used it to prove nonexistence results for closed PMC surfaces. The perturbation theory for PMC surfaces was studied. We obtained necessary conditions for the stability of PMC surfaces with planar boundaries.2.We posed a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we studied in detail a simple case where the solutions were vertical planar surfaces bounded by two vertical lines. We determined the stability or instability of each solution.3.We studied the geometry of surfaces which are in equilibrium for an anisotropic surface energy with a volume constraint. We obtained the first and second variations and studied the exceptional set of the Gauss map for such surfaces. Also we obtained representation formulas of the equilibrium surfaces of revolution and studied the geometry and stability of these surfaces.

1.研究了平均曲率为水平面以上高度的线性函数的曲面（PMC曲面）的变分理论。我们开发了一个通量公式，并用它来证明不存在的结果，关闭PMC曲面。研究了PMC曲面的摄动理论。我们得到了具有平面边界的PMC曲面稳定的必要条件。2.我们提出了一个曲面的变分问题，其解是一个由给定轮廓部分支撑的重力薄膜的几何模型。能量泛函包含表面张力、重力能和润湿能，欧拉-拉格朗日方程可以用表面的平均曲率、自由边界的曲率和其他一些几何量来表示。特别地，我们详细研究了一个简单的情况下，解决方案是垂直平面由两个垂直线界定。我们确定了每个解的稳定性或不稳定性。3.我们研究了具有体积约束的各向异性表面能的平衡表面的几何形状。我们得到了第一和第二变分，并研究了这类曲面的高斯映射的例外集。得到了旋转平衡曲面的表示公式，并研究了这些曲面的几何性质和稳定性。

项目成果

Miyuki Koiso: "Deformation and stability of surfaces with constant mean curvature"Tohoku Mathematical Journal. 54. 145-159 (2002)

Miyuki Koiso：“具有恒定平均曲率的曲面的变形和稳定性”东北数学杂志。

Miyuki Koiso: "Stability of surfaces with constant mean curvature in three-dimensional space forms"Differential Geometry, Valencia 2001-Proceedings of the International Conference Held to Honor the 60th Birthday of A.M.Naveira, Valencia, July 8-14, 2001(E

Miyuki Koiso：“三维空间形式中具有恒定平均曲率的表面的稳定性”微分几何，巴伦西亚 2001 年 - 为纪念 A.M.Naveira 60 岁生日而举行的国际会议记录，巴伦西亚，2001 年 7 月 8 日至 14 日（E）

Reiko Aiyama: "Lagrangian Surfaces with circle symmetry in the complex two-space"Michigan Mathematical Journal. 発行予定. (2004)

Reiko Aiyama：“复二空间中具有圆形对称性的拉格朗日曲面”，密歇根数学杂志即将出版（2004 年）。

Miyuki Koiso, Bennett Palmer: "Geometry and stability of bubbles with gravity"Indiana University Matheamtics Journal. (to appear).

Miyuki Koiso，Bennett Palmer：“重力气泡的几何结构和稳定性”印第安纳大学数学杂志。

Miyuki Koiso, Bennett Palmer: "On a variational problem for soap films with gravity and partially free boundary"Journal of the Mathematical Society of Japan. (to appear).

Miyuki Koiso、Bennett Palmer：“关于具有重力和部分自由边界的肥皂膜的变分问题”日本数学会杂志。