Rigidity and regularity of co-dimension one Sobolev isometric immersions

TL;DR

This paper establishes the developability and specific regularity of Sobolev isometric immersions of domains into Euclidean space, and shows they can be approximated by smooth isometries under certain conditions.

Contribution

It proves developability and $C^{1,1/2}$ regularity for $W^{2,2}$ Sobolev isometric immersions and demonstrates approximation by smooth isometries for convex domains.

Findings

Sobolev isometric immersions are developable and have $C^{1,1/2}$ regularity.

Such Sobolev isometries can be approximated by smooth isometries in the $W^{2,2}$ norm.

Results do not hold for weaker Sobolev regularities.

Abstract

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to be true if the Sobolev regularity is weaker than $W^{2,2}$.