Flat convergence for integral currents in metric spaces

By S. Wenger. 2005

70 citations recorded by the source corpus.

AbstractIn compact local Lipschitz neighborhood retracts in $$\mathbb{R}^n$$ weak convergence for integral currents is equivalent to convergence with respect to the flat distance. This comes as a consequence of the deformation theorem for currents in Euclidean space. Working in the setting of metric integral currents (the theory of which was developed by Ambrosio and Kirchheim) we prove that the equivalence of weak and flat convergence remains true in the more general context of metric spaces admitting local cone type inequalities. These include in particular all Banach spaces and all CAT(κ)-spaces. As an application we obtain the existence of a minimal element in a fixed homology class and show that the weak limit of a sequence of minimizers is itself a minimizer.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bb054426fe045f88e5034f25a5bb32fa5ba60d1f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Local Currents in Metric Spaces (2011, background): “The theory of metric currents has been further developed in [8, 31, 33], and some geometric applications have been found [32, 34].”
Area Minimizing Discs in Metric Spaces (2015, background): “In fact, it follows from [56] and [58] that the corollary holds for any geodesic metric space with the sharp constant 1 4π .”
The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces (2010, methodology): “Other important background to this paper is prior work of the second author, particularly [35], and a coauthored piece [32].”