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-Ampère 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 $p\ge \min \{2k,n\}$.

31 pages