Rigidity of codimension-1 isometric immersions in complete manifolds

TL;DR

This paper proves an asymptotic rigidity result for codimension-1 isometric immersions into complete manifolds, showing convergence of energy-vanishing sequences to isometric immersions using local rigidity estimates.

Contribution

It extends existing rigidity results to complete target manifolds, overcoming non-compactness issues with a novel local quantitative approach.

Findings

Sequences with vanishing elastic energy converge to isometric immersions.

The method avoids Young measures, offering a flexible analytical framework.

The approach applies to complete manifolds, broadening prior compactness assumptions.

Abstract

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.