Rectifiability of flat chains

Intrinsic rectifiability via flat cones in the Heisenberg group

We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones.

Rectifiability of flat chains in Banach spaces with coefficients in $Z_p$

Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and …

Sufficient condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $μ$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $μ$-almost all of $\text{supp}\,μ$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $α$ and $β_2$ numbers. The second one involves $α_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be neces…

Marstrand-Mattila rectifiability criterion for $1$-codimensional measures in Carnot Groups

This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups $\mathbb{H}^n$ are $C^1_{\mathbb{H}^n}$-rectifiable, providing the first non-Euclidean extension of Preiss's rectifiability theorem and a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups.

小常数一致可求长性

Small constant uniform rectifiability

本文研究小常数条件下的一致可求长集合，证明局部平坦、Ahlfors 正则且密度接近 1 的集合满足 Tolsa α 系数的小常数 Carleson 测度估计，从而给出弦弧域的新等价刻画。

Necessary condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $μ$ is $n$-rectifiable if $μ\ll\mathcal{H}^n$ and $μ$-almost all of $\text{supp}\,μ$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give a necessary condition for rectifiability in terms of the so-called $α_2$ numbers -- coefficients quantifying flatness using Wasserstein distance $W_2$. In a recent article we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.