Asymptotic rigidity for shells in non-Euclidean elasticity

TL;DR

This paper proves that in non-Euclidean elasticity, shells with nearly minimal energy converge to isometric immersions with prescribed fundamental forms, extending rigidity results to curved ambient spaces.

Contribution

It establishes an asymptotic rigidity theorem for shells modeled as Riemannian manifolds immersed in constant curvature spaces, generalizing Reshetnyak's theorem.

Findings

Sequences of low-energy immersions converge to isometric immersions with reference fundamental forms.

In Euclidean space, the fundamental forms satisfy Gauss-Codazzi-Mainardi conditions.

The result extends rigidity theorems to shells in curved ambient spaces.

Abstract

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