Asymptotic rigidity for shells in non-Euclidean elasticity

We consider a prototypical "stretching plus bending" functional of an elastic shell. The shell is modeled as d-dimensional Riemannian manifold endowed, in addition to the metric, with reference second fundamental form. immersed into (d+1)-dimensional ambient space, and energy accounts for deviations induced metric forms from their values. Under assumption that space constant sectional curvature, we prove any sequence immersions asymptotically vanishing converges isometric immersion having In particular, if Euclidean then form satisfy Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed (manifold-valued) co-dimension 1 analog Reshetnyak's asymptotic rigidity theorem. It also relates recent results on continuity surfaces respect forms.