Decomposition of acyclic normal currents in a metric space

Abstract

We prove that every acyclic normal one-dimensional real Ambrosio–Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space under a suitable set-theoretic assumption

Three superposition principles: Currents, continuity equations and curves of measures

“…Here we slightly strengthen [25, theorem 6.1] relating it to the superposition principle, whenever the underlying space E is a Banach space with Radon-Nikodym property and strictly convex norm, by proving that any measure on curves provided by the latter for the continuity equation satisfied by the naturally defined flow of µ, in fact represents the same vector field of T , or, in other words, represents a normal current in space with the same underlying vector field as T and with different mass measure (still absolutely continuous with respect to that of T ). Let us stress that our proof technique is based on the more recent approach of [24], ultimately relying on a reduction to polygonal currents, and not on Smirnov's original proof. In fact, in view of Smirnov's original argument, it seems reasonable that it should allow one to deduce a superposition results for currents from that for continuity equations, reversing our main implication (of course, one should first establish a general result for continuity equations by other means).…”

“…The results of type (C) for De Rham currents have been proven first by S. Smirnov [28] (later several different proofs have been given for partial results of this kind, see e.g. [27] and references therein, and also [16] for an interesting discrete analogue), and for general metric current in [24,26]. In this paper we show in fact that the result on representation of acyclic metric currents provides (A) for general metric spaces.…”

Metric currents and the Poincaré inequality

“…Then 13) and ψ x and is (C/r n )-Lipschitz on B 0 : this is easy to check, noting that Proof. If u(y) = 1, then…”

“…By [13, Theorem 5.1], a normal acyclic 1-current T on X is decomposable in arcs. Combining [13,Definition 4.4] and [13,Lemma 4.17], this means that there exists a transport η on Γ such that η almost every curve in Γ is an arc, and moreover the following equalities hold: Now, we apply the decomposition result to our concrete situation. Assume that s, t ∈ X, and T is a GPC joining s to t, as in Definition 1.3.…”

“…= (∂T ) + and η(2) = (∂T ) − . (6.2)If the reader checks carefully[13, Lemma 4.17], then she will find the measure "µ[[θ]] " (in our notation [[θ]] ) in place of "H 1 | θ " in (6.1), but these measures coincide if θ is an arc. Indeed, in that case[14, Lemma 2.2] shows that µ [[θ]] = θ ♯ (|θ|L 1 ), where |θ| denotes the metric derivative.…”