Thin Structures With Imposed Metric

We consider thin structures with a non necessarily realizable imposed metric, that only depends on the surface variable. We give a unified presentation of the three main limit models. We establish the generalized membrane model and we show, by means of an algebraic proof, that the internal membrane energy vanishes on short maps of the metric restricted to the plane. We recall that a generalized bending model can occur only when this reduced metric admits sufficiently regular isometric immersions. When the entries R12.. of the Riemannian curvature tensor are null, this bending energy can vanish; then the next model is necessarily a generalized von Kármán model whose minimum is zero if and only if the three-dimensional metric is flat