Abstract
In 1841 the astronomer/mathematician C. Delaunay isolated a certain class of surfaces in Euclidean space, representations of which he described explicitly [1]. In an appendix to that paper, M. Sturm characterized Delaunay’s surfaces variationally; indeed, as the solutions to an isoperimetric problem in the calculus of variations. That in turn revealed how those surfaces make their appearance in gas dynamics; soap bubbles and stems of plants provide simple examples. See Chapter V of the marvellous book [8] by D’Arcy Thompson for an essay on the occurrence and properties of such surfaces in nature.
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References
C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante. J. Math. pures et appl. Sér. 1(6) (1841), 309–320. With a note appended by M. Sturm.
J. Eells, On the surfaces of Delaunay and their Gauss maps. Proc. IV Int. Colloq. Diff. Geo. Santiago de Compostela (1978), 97–116.
J. Eells and L. Lemaire, A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1–68.
J. Eells and L. Lemaire, On the construction of harmonic and holomorphic maps between surfaces. Math. Ann. 252 (1980), 27–52.
A. R. Forsyth, Calculus of variations. Cambridge (1927).
E. A. Ruh and J. Vilms, The tension field of the Gauss map. Trans. Amer. Math. Soc. 149 (1970), 569–573.
R. T. Smith, Harmonic mappings of spheres. University of Warwick Thesis (1972).
D’A. W. Thompson, Growth and form. Cambridge (1917).
C. Zwikker, The advanced geometry of plane curves and their applications. Dover (1963).
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Eells, J. (2001). The Surfaces of Delaunay. In: Mathematical Conversations. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0195-0_14
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DOI: https://doi.org/10.1007/978-1-4613-0195-0_14
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