Stability of isometric immersions of hypersurfaces

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to L^p-perturbations of their fundamental forms: For a manifold M^d endowed with a reference metric and a reference shape operator, we show that a sequence of immersions f_n : M^d → N^d+¹, whose pullback metrics and shape operators are arbitrary close in L^p to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold N, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.