In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

1962年，Wunderlich发表了文章“关于可发展的莫比乌斯带”，其中他试图确定自由站立的莫比乌斯带的平衡形状。与萨多斯基在无穷小莫比乌斯带上的开创性工作相一致，Wunderlich使用了能量最小化原理，该原理断言，莫比乌斯带的平衡形状在所有可能的形状中具有最低的弯曲能。通过利用带的可显影性，Wunderlich在不假定带的宽度较小的情况下将弯曲能从表面积分减小为线积分。尽管Wunderlich未能完全成功地确定Möbius谱带的平衡形状，但他在尺寸上减小的能量积分可以说是该领域最重要的发展之一。在这项工作中 我们严格地证明了Wunderlich积分的有效性，并充分阐述了能量最小化问题，该问题与找到闭合带的平衡形状有关，包括任意数量的扭曲的可定向和不可定向带。这包括表征能量函数的函数空间，处理等轴测图和局部注入性约束，以及推导Euler-Lagrange方程。应特别注意连接边缘条件，变形带的规律性，确定确保变形是排斥性所需的参数空间，减少等轴测约束以及推导与Euler-Lagrange方程相关的匹配条件和跳跃条件。