A geometric rigidity estimate for codimension-1 immersions into spheres

We establish a quantitative stability estimate for codimension-one immersions into a round sphere. Let $(M,g)$ be an oriented Riemannian manifold and suppose that a prescribed shape operator is realised by a smooth isometric immersion $\theta :(M,g)\to {\mathbb{S}}^{n+1}$. We prove that, on every relatively compact strongly Lipschitz domain and for every $1<p<\mathrm{\infty }$, any Sobolev immersion $\phi$ is close to $\theta$ modulo an ambient rotation, with the $W^{1,p}$-distances between both the immersions and their Gauss maps controlled by the $L^p$ stretching-plus-bending energies of $\phi$. No {\it a priori} bounds on the fundamental forms of $\phi$ are required. The proof proceeds by extending the immersions along normal geodesics and reducing the problem to an equidimensional geometric rigidity estimate on the sphere. A finite localisation and patching argument handles the possible non-injectivity of the normal extension of the reference immersion.