Abstract
In 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.
Similar content being viewed by others
References
Ambrosio L., Kirchheim B. (2000). Currents in metric spaces. Acta Math. 185(1):1–80
Ballmann W. (1995). Lectures on spaces of nonpositive curvature. DMV Seminar Band 25, Birkhäuser, Basel
Burago, D., Burago, Yu., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33, Amer. Math. Soc., Providence, Rhode Island (2001)
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Grundlehren der mathematischen Wissenschaften 319. Springer, Berlin Heidelberg New York (1999)
De Giorgi E. (1995). Problema di Plateau generale e funzionali geodetici. Atti Sem Mat. Fis. Univ. Modena 43:282–292
Eilenberg S., Steenrod N. (1952). Foundations of algebraic topology. Princeton University Press, Princeton
Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47:324–353
Federer, H.: Geometric Measure Theory. Springer Berlin Heidelberg New York (1969,1996)
Federer H., Fleming W.H. (1960). Normal and integral currents. Ann. Math. 72:458–520
Kirchheim B. (1994). Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1):113–123
Wenger S. (2005). Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2):534–554
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio
Rights and permissions
About this article
Cite this article
Wenger, S. Flat convergence for integral currents in metric spaces. Calc. Var. 28, 139–160 (2007). https://doi.org/10.1007/s00526-006-0034-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-006-0034-0