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

TL;DR

The paper resolves Lang's Flat Chain Conjecture by revealing a deep link between flat chains and the regularity of the prescribed Jacobian equation near $L^\infty$. It shows strong Lipschitz non-regularity for the equation $\sum_i f_i \det \mathrm{D}\pi_i = \rho$ and uses a Hahn–Banach separation framework to prove non-surjectivity of the associated embedding, which implies Lang's conjecture fails for all $d\ge 2$ and $k\in\{1,\dots,d\}$. Consequently, a complete classification is obtained: the Flat Chain Conjecture of Lang holds only in the trivial cases $k=0$ or $k=d=1$, while the Ambrosio–Kirchheim FCC remains open in general. The exterior derivative case is also shown to lack Lipschitz regularity, generalizing the non-regularity phenomenon to a broader PDE setting with implications for geometric measure theory in metric spaces.

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.

Figures (5)

• Figure 1: A diagram indicating the relationship between function spaces and potential spaces. The dashed arrows only indicate direction of duality. The full arrows correspond to operators. The bottom part of the diagram commutes.
• Figure 2: In the picture, $y = x+\frac{c}{N}e_1$ and $p=z+ \frac{c}{NM}e_1$. The statement of Lemma \ref{['L:dichotomy']} is that if the stretching of $h$ on all pairs of points $z$ and $p$ is controlled by $(1+\varphi)$ times the stretching of $h$ on the endpoints $0$ and $ce_1$ (i.e. statement \ref{['Enum:DKK2']} does not hold) then there must be some $Q_i$ such that $h(x)-h(y)\approx \textnormal{const}$ for all pairs $x, y$ where $y=x+\frac{1}{N}e_1$ and $x\in Q_i$. If this is the case, this in particular implies that the image, under $h$, of $Q_{i+1}$ is nearly a translation of the image of $Q_i$.
• Figure 3: A picture of a cube for $d=2$, $M=5$ and $K=7$. The denoted rectangle $R$ (in red) is the rectangle $R(Q,(\frac{2}{M},\frac{1}{M}))$. Note that for any $z\in\textnormal{Ref}$, $R(Q,z)\subset Q$ and moreover, if $K$ is very large, then the measure of the set $Q\setminus \bigcup_z R(Q,z)$ is nearly the measure of $Q$. In the case $d=2$, we have explicitly $\mathcal{H}^2(Q\setminus \bigcup_z R(Q,z))=\mathcal{H}^2(Q)(1-\frac{1}{K}) = r^2 - \frac{r^2}{K}$. For each of the indicated rectangles on the picture, their natural partition into subcubes is also indicated, cf. \ref{['E:partitioning-rect-into-cubes']}. For example, $R=\bigcup_{i=1}^7 Q(R,i)$, where each $Q(R,i)$ is one of the very small squares inside the red rectangle $R$. If $Q=Q_{i_0, \dots, i_k}^{z_1, \dots, z_k}$, then each of the small squares is of the form $Q_{i_0, \dots, i_k, i_{k+1}}^{z_1, \dots, z_{k+1}}$ and $R=R_{i_0, \dots, i_k}^{z_1, \dots, z_k, {(\frac{2}{M},\frac{1}{M})}}$.
• Figure 4: A picture of a rectangle together with its lattice, in the case $d=2$, $K=7$ and $M=5$. If $R=R_{i_0, i_1, \dots, i_{k-1}}^{z_1,\dots, z_{k}}$, then the part of lattice in blue is a rescaled and translated copy of the reference lattice $\textnormal{Ref}$. Explicitly, it is equal to $\frac{1}{K}\frac{1}{(KM)^k}\textnormal{Ref}+p(R_{i_0, i_1, \dots, i_{k-1}}^{z_1,\dots, z_{k}}) + (0, \frac{3}{K(KM)^k})$. The $3$ appears because the blue lattice lies in the fourth square from the left and the indexing of the squares starts with $0$. A different way of writing the blue lattice is $\frac{1}{K}\frac{1}{(KM)^k}\textnormal{Ref}+p(Q_{i_0, i_1, \dots, i_{k-1}, 3}^{z_1,\dots, z_{k}})$. We further remark that for any pair of horizontally adjacent lattice points, say $p,q$, with $p$ left of $q$, there is a rectangle of the form $R_{i_0, i_1, \dots, i_{k}}^{z_1,\dots, z_{k+1}}$ such that $p$ is the left endpoint of $R_{i_0, i_1, \dots, i_{k}}^{z_1,\dots, z_{k+1}}$ while $q$ is the right endpoint of $R_{i_0, i_1, \dots, i_{k}}^{z_1,\dots, z_{k+1}}$. The tiny red rectangle is the rectangle $R_{i_0, i_1, \dots, i_{k-1}, 3}^{z_1,\dots, z_{k}, (2,3)}$.
• Figure 5: Picture of $\operatorname{refine}_1(\rho)$ in case $K=6$, $M=4$ and $d=2$. The whole rectangle is the initial rectangle $R_0$. The black regions correspond to sets on which $\operatorname{refine}_1(\rho)=1$ and the white regions correspond to sets on which $\operatorname{refine}_1(\rho)=2$.

Theorems & Definitions (115)

• Conjecture 1.1
• Definition 1.2: Lipschitz solutions
• Definition 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3