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In line with Sadowsky\u2019s pioneering works on M\u00f6bius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the M\u00f6bius 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 M\u00f6bius 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\u2013Lagrange 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\u2013Lagrange equations.<\/jats:p>","DOI":"10.1007\/s00332-020-09630-z","type":"journal-article","created":{"date-parts":[[2020,5,22]],"date-time":"2020-05-22T18:27:30Z","timestamp":1590172050000},"page":"2577-2611","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Closed Unstretchable Knotless Ribbons and the Wunderlich Functional"],"prefix":"10.1007","volume":"30","author":[{"given":"Brian","family":"Seguin","sequence":"first","affiliation":[]},{"given":"Yi-chao","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Eliot","family":"Fried","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,5,22]]},"reference":[{"key":"9630_CR1","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s10659-014-9501-6","volume":"119","author":"S Bartels","year":"2015","unstructured":"Bartels, S., Hornung, P.: Bending paper and the M\u00f6bius strip. 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