On a nonlinear elastic shell system in liquid crystal theory: generalized Willmore surfaces and Dupin cyclides.
Résumé
An elastic membrane model of smectic A liquid crystal deformation is derived ab initio via a variational approach. The well-determined nature of the resulting nonlinear model equations reveals that the deformed states of the liquid crystal lamellae can only adopt privileged geometries. These are shown to generalize classical and novel ‘integrable' geometries associated with Willmore, linear Weingarten and ‘membrane' O surfaces. The main result establishes that, remarkably, the membrane model admits layered parallel Dupin cyclide structures of the kind originally observed by Friedel and Grandjean in their pioneering experiments of 1910 and subsequently elaborated upon by Friedel in 1922 and later by Bragg.