Flat convergence for integral currents in metric spaces



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.


Integral currents Flat distance Weak convergence Alexandrov spaces 


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  1. 1.
    Ambrosio L., Kirchheim B. (2000). Currents in metric spaces. Acta Math. 185(1):1–80MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ballmann W. (1995). Lectures on spaces of nonpositive curvature. DMV Seminar Band 25, Birkhäuser, BaselMATHGoogle Scholar
  3. 3.
    Burago, D., Burago, Yu., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics, vol. 33, Amer. Math. Soc., Providence, Rhode Island (2001)Google Scholar
  4. 4.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Grundlehren der mathematischen Wissenschaften 319. Springer, Berlin Heidelberg New York (1999)Google Scholar
  5. 5.
    De Giorgi E. (1995). Problema di Plateau generale e funzionali geodetici. Atti Sem Mat. Fis. Univ. Modena 43:282–292MathSciNetGoogle Scholar
  6. 6.
    Eilenberg S., Steenrod N. (1952). Foundations of algebraic topology. Princeton University Press, PrincetonMATHGoogle Scholar
  7. 7.
    Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47:324–353MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Federer, H.: Geometric Measure Theory. Springer Berlin Heidelberg New York (1969,1996)Google Scholar
  9. 9.
    Federer H., Fleming W.H. (1960). Normal and integral currents. Ann. Math. 72:458–520MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kirchheim B. (1994). Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1):113–123MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Wenger S. (2005). Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2):534–554MATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA

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