A variational analysis of nematic axisymmetric films: the covariant derivative case

Nematic surfaces are thin fluid structures, ideally two-dimensional, endowed with an in-plane nematic order. In 2012, two variational models have been introduced by Giomi \cite{giomi2012hyperbolic} and by Napoli and Vergori \cite{napoli2012surface, napoli2018influence}. Both penalize the area of the surface and the gradient of the director: in \cite{giomi2012hyperbolic} the covariant derivative of the director is considered, while \cite{napoli2018influence} deals with the surface gradient. In this paper, a complete variational analysis of the model proposed by Giomi is performed for revolution surfaces spanning two coaxial rings.