Flat chains of finite size in metric spaces

Abstract

In this paper we investigate the notion of flat current in the metric spaces setting, and in particular we provide a definition of size of a flat current with possibly infinite mass. Exploiting the special nature of the 0-dimensional slices and the theory of metric-space valued BV functions we prove that a k-current with finite size T sits on a countably ${\mathcal{H}}^k$-rectifiable set, denoted by $\mathrm{set}(T)$. Moreover we relate the size measure of T to the geometry of the tangent space ${\mathrm{Tan}}^{(k)}(\mathrm{set}(T),x)$.