Flat convergence for integral currents in metric spaces. (English) Zbl 1110.53030
Summary: 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 \(\text{CAT}_{(\kappa)}\)-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.
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 58A25 | Currents in global analysis |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
References:
| [1] | Ambrosio L., Kirchheim B. (2000). Currents in metric spaces. Acta Math. 185(1):1–80 · Zbl 0984.49025 · doi:10.1007/BF02392711 |
| [2] | Ballmann W. (1995). Lectures on spaces of nonpositive curvature. DMV Seminar Band 25, Birkhäuser, Basel · Zbl 0834.53003 |
| [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) · Zbl 0981.51016 |
| [4] | Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Grundlehren der mathematischen Wissenschaften 319. Springer, Berlin Heidelberg New York (1999) |
| [5] | De Giorgi E. (1995). Problema di Plateau generale e funzionali geodetici. Atti Sem Mat. Fis. Univ. Modena 43:282–292 |
| [6] | Eilenberg S., Steenrod N. (1952). Foundations of algebraic topology. Princeton University Press, Princeton · Zbl 0047.41402 |
| [7] | Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47:324–353 · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
| [8] | Federer, H.: Geometric Measure Theory. Springer Berlin Heidelberg New York (1969,1996) · Zbl 0176.00801 |
| [9] | Federer H., Fleming W.H. (1960). Normal and integral currents. Ann. Math. 72:458–520 · Zbl 0187.31301 · doi:10.2307/1970227 |
| [10] | Kirchheim B. (1994). Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1):113–123 · Zbl 0806.28004 · doi:10.1090/S0002-9939-1994-1189747-7 |
| [11] | Wenger S. (2005). Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2):534–554 · Zbl 1084.53037 · doi:10.1007/s00039-005-0515-x |
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