Sobolev spaces of isometric immersions of arbitrary dimension and codimension

Abstract: We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min{2k, n}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Amp`ere equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1)\times (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min{2k, n}$.