On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $\Bbb R^D$

TL;DR

This paper investigates conditions under which a small distortion map defined on a finite subset of Euclidean space can be extended to a smooth, small distortion map on the entire space, addressing a fundamental problem in geometric analysis.

Contribution

It provides a solution to the Whitney extension-interpolation-alignment problem for small distortions in Euclidean spaces, establishing when such extensions are possible.

Findings

Characterization of extendability conditions for small distortions

Construction of smooth extensions preserving small distortion

Resolution of the extension problem in Euclidean spaces for finite sets

Abstract

In this paper, we study the following problem. Let $D\geq 2$, $S\subset \mathbb R^D$ be finite and let $\phi:S\to \mathbb R^D$ with $\phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $\phi$ can be extended to a function $\Phi:\mathbb R^D\to \mathbb R^D$ which is a smooth small distortion on $\mathbb R^D$. The work in this paper appears in the research memoir [14]