Smooth approximation of mappings with rank of the derivative at most $1$

Paweł Goldstein, Piotr Hajłasz

It was conjectured that if $f\in C^1({\mathbb{R}}^n,{\mathbb{R}}^n)$ satisfies $\mathrm{rank}Df\le m<n$ everywhere in ${\mathbb{R}}^n$, then $f$ can be uniformly approximated by…