We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.

中文翻译：

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在外力作用下具有边缘的轴对称膜：屈曲、最小表面和系绳

我们使用理论和数值计算来确定具有抗弯曲和恒定面积的轴对称流体膜的形状。该膜以经典几何形状连接两个环，在肥皂膜中产生悬链线形状。在我们的问题中，我们找到了形状和外力作为环分离函数的无限多分支解，类似于细杆欧拉屈曲的无限本征模式族。特别注意悬链线，当面积小于等于两个环包围的平面面积的临界面积时，它表现为最大允许分离的形状。扰动理论的论点直接将悬链膜的张力与此状态下悬链皂膜的稳定性联系起来。当膜面积大于临界面积时，我们发现大环分离时形状方程的额外圆柱系链解，并且任意大的环分离都是可能的。这些结果适用于消失的高斯曲率模量的情况；当高斯曲率模量非零且面积低于临界面积时，力和膜张力随着环分离接近其最大值而发散。我们还检查了我们形状的稳定性，并通过分析表明，悬链膜的稳定性特性与其肥皂膜对应物明显不同。这些结果适用于消失的高斯曲率模量的情况；当高斯曲率模量非零且面积低于临界面积时，力和膜张力随着环分离接近其最大值而发散。我们还检查了我们形状的稳定性，并通过分析表明，悬链膜的稳定性特性与其肥皂膜对应物明显不同。这些结果适用于消失的高斯曲率模量的情况；当高斯曲率模量非零且面积低于临界面积时，力和膜张力随着环分离接近其最大值而发散。我们还检查了我们形状的稳定性，并通过分析表明，悬链膜的稳定性特性与其肥皂膜对应物明显不同。