Rigidity and flexibility of isometric extensions

Abstract

In this paper we consider the rigidity and flexibility of $C^{1,\theta }$ isometric extensions. We show that the Hölder exponent ${\theta }_0=\frac{1}{2}$ 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 $\theta >\frac{1}{2}$, then the tangential connection agrees with the Levi-Civita connection along $\Sigma$. On the other hand, for any $\theta <\frac{1}{2}$ 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, $\theta <\frac{1}{2}$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.