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Crystallographic aspects of the Bonnet transformation for periodic minimal surfaces (and crystals of films)

(Received 2 July 1992, accepted in final form 29 October 1992)

Abstract
The surfaces with vanishing mean curvature, or minimal surfaces, are linear projections of minimal surfaces imbedded in the complex 3 dimensional space C ³. For real minimal surfaces, the Bonnet transformations form a one parameter group of isometries which correspond to generalised rotations in the complex or higher dimensional space. The translation symmetries of fully periodic minimal surfaces in both R ³ and C ³ are investigated, with emphasis on the example of the P, D and G surfaces. The relevance of the Bonnet transformation for physical transitions is then discussed in the light of higher dimensional crystallography.