Rigidity of codimension-1 isometric immersions in complete manifolds

We establish an asymptotic rigidity result for isometric immersions of codimension-1. Specifically, we consider a sequence of immersions from a compact $d$-dimensional manifold into a complete $(d+1)$-dimensional manifold whose elastic energies vanish asymptotically, where the elastic energy quantifies both stretching and bending. We show that such a sequence admits a subsequence converging to an isometric immersion. This extends a result of Alpern, Kupferman, and Maor to the case of complete target manifolds, where the lack of compactness introduces additional analytical difficulties. The proof is based on an approach using local quantitative rigidity estimates, obtained via a reduction to the Euclidean setting. This method avoids the use of Young measures and provides a flexible framework that may be of independent interest.