Rigidity and Flexibility of Isometric Extensions

In this paper we consider the rigidity and flexibility of $C^{1, \theta}$ isometric extensions and we show that the H\"older exponent $\theta_0=\frac12$ is critical in the following sense: if $u\in C^{1,\theta}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $\Sigma$ and $\theta> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $\Sigma$. On the other hand, for any $\theta<\frac12$ we can construct $C^{1,\theta}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, \theta}$ isometric embeddings, $\theta<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.