$C^1$ 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

TL;DR

This paper constructs a specific $C^1$ mapping in $R^5$ with rank at most 3 that cannot be uniformly approximated by smoother $C^2$ mappings with the same rank constraint, countering a previous conjecture.

Contribution

It provides a counterexample to G{ {a}}l{}ski's conjecture, showing limitations in approximating certain low-rank $C^1$ mappings by $C^2$ mappings.

Findings

Counterexample to G{ {a}}l{}ski's conjecture

Limitations on uniform approximation of low-rank $C^1$ maps

Implications for the theory of rank-constrained mappings

Abstract

We find a counterexample to a conjecture of Ga{\l}\k{e}ski by constructing for some positive integers $m<n$ a mapping $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfying $\mathrm{rank}\, Df\leq m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_\varepsilon$ satisfying the same rank constraint $\mathrm{rank}\, Df_\varepsilon\leq m$.