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

TL;DR

This paper demonstrates the failure of Lang's Flat Chain Conjecture for certain metric currents in higher dimensions by linking it to the non-regularity of the prescribed Jacobian equation, providing new insights into geometric measure theory.

Contribution

It proves the conjecture fails in specific cases and establishes the non-regularity of solutions to the prescribed Jacobian equation near $L^ ablafty$, advancing understanding of geometric measure theory.

Findings

Lang's Flat Chain Conjecture fails for $d\\geq 2$ and $k\\in\{1, \\dots, d\}$

The prescribed Jacobian equation lacks the necessary regularity near $L^ ablafty$

Quantifies the smallness of the convex hull of determinants of derivatives of Lipschitz vector fields

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 $\pi$, 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} \pi: \operatorname{Lip}(\pi)\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.