On the Whitney distortion extension problem for $C^m({\mathbb{R}}^n)$ and $C^{\mathrm{\infty }}({\mathbb{R}}^n)$ and its applications to interpolation and alignment of data in ${\mathbb{R}}^n$

In this announcement we consider the following problem. Let $n,m\ge 1$, $U\subset {\mathbb{R}}^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$bea$C^m$map\mathrm{.}If$E\subset U$ is compact (with some geometry) and the restriction of $\varphi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m({\mathbb{R}}^n)$ one-to-one and onto almost isometry $\mathrm{\Phi }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with small distortion which agrees with $\varphi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$awayfrom$E$\mathrm{.}(b)Let$\phi:U\to \mathbb R^n$be$C^{\infty}$map\mathrm{.}If$E\subset U$iscompact(withsomegeometry)andtherestrictionof$\phi$to$E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\mathrm{\infty }}({\mathbb{R}}^n)$ one-to-one and onto almost isometry $\mathrm{\Phi }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ with small distortion which agrees with $\varphi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$awayfrom$E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in ${\mathbb{R}}^n$. The material in this paper appears in the memoir [14].