preprint

Inserted: 3 mar 2026

Year: 2025

Abstract:

We show that Lang's Flat Chain Conjecture (that is, without requiring finite mass of the underlying currents) fails for metric $k$-currents in $\mathbb{R}^d$ whenever $d\geq 2$ and $k\in\{1, \dots, d\}$. In all other cases, it holds. The original conjecture due to Ambrosio and Kirchheim remains open. We first connect Lang's conjecture to a regularity statement concerning the prescribed Jacobian equation near $L^\infty$. We then show that the equation does not have the required regularity. For a Lipschitz vector field $π$, its derivative $\mathrm{D}π$ exists a.e. and is identified with a matrix. Our non-regularity results for the prescribed Jacobian equation quantify how "small" the set \begin{equation} \operatorname{conv}(\{\operatorname{det}\mathrm{D} π: \operatorname{Lip}(π)\leq L\})\subset L^\infty \end{equation} is for every $L>0$. The symbol "$\operatorname{conv}$" stands for the convex hull. The "smallness" is quantified in topological terms and is used to show that Lang's Flat Chain Conjecture fails.