$C^1$ mappings in ${\mathbb{R}}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3

Abstract

We find a counterexample to a conjecture of Gałęski by constructing for some positive integers $m<n$ a mapping $f\in C^1({\mathbb{R}}^n,{\mathbb{R}}^n)$ satisfying $\mathrm{rank}Df\le m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_{\epsilon }$ satisfying the same rank constraint $\mathrm{rank}D{f}_{\epsilon }\le m$.