7 problems

Let $k\in\mathbb{N}$, let $\mathrm{X}$ be a metric space, and let $\mathbf{M}_k(\mathrm{X})$ denote the space of metric $k$-currents on $\mathrm{X}$. Normal $k$-currents are metric…

Lang's general Flat Chain Conjecture. For every $d\in\mathbb{N}$ and $k\in\mathbb{N}\cup\{0\}$, if $T\in\mathcal{D}_k(\boldsymbol{R}^d)$ has compact support, then

Finite-flat-norm variant of Lang's conjecture. Every compactly supported metric $k$-current in $\boldsymbol{R}^d$ with finite flat norm is a flat chain.

Top-dimensional Flat Chain Conjecture. There exists some $u\in L^1(Q)$ such that

Lang's Flat Chain Conjecture. Every compactly supported metric $k$-current in $\boldsymbol{R}^d$ is a flat $k$-chain.

Let $m$-dimensional $(\alpha_0,\dots,\alpha_m)$-Hölder currents in $\mathbf{R}^d$ be the multilinear currents defined using Hölder functions, and set … When…

Let $T$ be a metric $k$-current on Euclidean space, and let $\widetilde{T}$ denote the associated flat chain. The map sending a flat chain with finite mass to the metric $k$-curren…