Uniform $W^{2,m}$ approximation of almost isometric maps

Abstract

This paper shows that an almost-isometric Sobolev map from a Lipschitz subset of an oriented manifold $M$ into another oriented manifold $N$ can be approximated by a map of class $C^{1,\alpha }$, with a universal bound on the $W^{2,m}$ norm, for any $m\in (1,\infty )$. This, in turn, implies a universal bound on the $C^{1,\alpha }$ seminorm. The $W^{1,p}$ distance of the approximation from the original map is controlled in terms of the $L^p$-deviation from being an isometry, with the optimal scaling. The manifolds $M$ and $N$ are assumed to be compact, oriented and have equal dimension; $M$ may have a boundary.