Failure of Lang's Flat Chain Conjecture and non-regularity of the prescribed Jacobian equation

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 $d\ge 2$ 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 $L^{\mathrm{\infty }}$. We then show that the equation does not have the required regularity. For a Lipschitz vector field $\pi$, its derivative $\mathrm{D}\pi$ 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 “$\mathrm{conv}$” stands for the convex hull. The “smallness” is quantified in topological terms and is used to show that Lang’s Flat Chain Conjecture fails.