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

Abstract

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 $C^{\infty }$-mappings $g$ satisfying $\mathrm{rank}Dg\le m$ everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when $m=1$. More precisely, if $m=1$, our result yields an almost-uniform approximation of locally Lipschitz mappings $f:\Omega \to {\mathbb{R}}^n$, satisfying $\mathrm{rank}Df\le 1$ a.e., by $C^{\infty }$-mappings $g$ with $\mathrm{rank}Dg\le 1$, provided $\Omega \subset {\mathbb{R}}^n$ is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ($\mathbb{R}$-trees).