| Gaia: | 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
${\mathcal M}^d$
endowed with a reference metric and a reference shape operator, we show that a sequence of immersions
$f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$
, 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
${\mathcal N}$
, 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.
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