Abstract
We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we deduce a generalization of the chain rule, and show that currents of absolutely continuous mass are given by integration against measurable k-vector fields. We further prove that if the underlying metric space is a Carnot group with its Carnot-Caratheodory distance, then every metric current T satisfies T !θ= 0 and T !dθ= 0, whenever θ ∈ 1(G) annihilates the horizontal bundle of G. Moreover, this condition is necessary and sufficient for a metric current with respect to the Riemannian metric to extend to one with respect to the Carnot-Caratheodory metric, provided the current either is locally normal, or has absolutely continuous mass. Mathematics Subject Classification (2010): 30L99 (primary); 49Q15 (secondary).
Highlights
In [1], Ambrosio and Kirchheim introduced a definition of currents in metric spaces, extending the theory of normal and integral currents developed by Federer and Fleming [10] for Euclidean spaces
The extension of these classes of currents allows the formulation of variational problems in metric spaces, and the validity of the compactness and closure theorems of [10], proven in the metric setting in [1], allows for their solution
In this paper we investigate the theory of metric currents in spaces that admit differentiable structures, in the sense of Cheeger [4] and Keith [18], with a particular emphasis on Carnot Groups equipped with their Carnot-Caratheodory metrics
Summary
In [1], Ambrosio and Kirchheim introduced a definition of currents in metric spaces, extending the theory of normal and integral currents developed by Federer and Fleming [10] for Euclidean spaces. A metric k-current T ∈ Mk(X ) is defined to be a real-valued function on Dk(X) that is linear in each argument, continuous in an appropriate sense, vanishes where it ought to (namely, on forms f dg1 ∧ · · · ∧ dgk such that one of the functions gi is constant on the support of f ), and satisfies a finite mass condition They demonstrated that most of the results of [10] carry over to this more general setting, and that the classes of classical and metric normal currents are naturally isomorphic in the Euclidean case.
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