Asymptotic rigidity of codimension-1 isometric immersions via quantitative estimates

We offer an alternative approach to the asymptotic rigidity of codimension-1 isometric immersions via quantitative rigidity estimates. We show that an immersion between compact manifolds $M$ and $N$ of dimensions $d$ and $d + 1$, respectively, with small stretching plus bending energy is close to an isometric immersion. In this way, we recover the results of Alpern, Kupferman, and Maor. In contrast to their intrinsic approach, we reduce the problem to the equidimensional Euclidean setting and apply the Friesecke-James-M\"uller rigidity estimate to obtain quantitative results. This yields an elementary proof based on Euclidean techniques. The rigidity estimates are of independent interest.