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 $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank} Dg\leq m$ 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 $\operatorname{rank} Df\leq 1$ a.e., by $C^\infty$-mappings $g$ with $\operatorname{rank} Dg\leq 1$, 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).