On Extensions of e Diffeomorphisms
Year: 2015
Type: preprint
Abstract: Let $n\geq 1$, $U\subset\mathbb R^n$ open and $\phi:U\to \mathbb R^n$ be $C^1$ map. If $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry, we study the question of how to decide when $\phi$ extends to a smooth distorted isometry $\Phi:\mathbb R^n\to \mathbb R^n$ and we study how well $\Phi$ can be approximated by Euclidean motions on $\mathbb R^n$. These results depend on relations between the distortions of $\phi$ and $\Phi$, the geometry of $E$ and the dimension $n$. Our results complement those of [15,16] which deal with the case when $E$ is a finite set. There, the results depend on relations between the distortions of $\phi$ and $\Phi$, the cardinality of $E$, the distribution of the points of $E$ and the dimension $n$. When $E$ is finite, the problem above is that of interpolation and alignment.
Source: arXiv (Cornell University)
Authors: Steven B. Damelin, Charles Fefferman
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2162507177