Abstract
By Gromov’s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio–Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio L., Kirchheim B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000)
Bridson M., Haefliger A.: Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften 319. Springer-Verlag, Berlin (1999)
Cheeger J., Colding T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)
Cheeger J., Colding T.H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54(1), 13–35 (2000)
Cheeger J., Colding T.H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)
Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Federer H., Fleming W.H.: Normal and integral currents. Ann. Math. (2) 72, 458–520 (1960)
Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Publ. Math. No. 53, pp. 53–73 (1981)
Gromov M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian spaces, with Appendices by Katz, M., Pansu, P., Semmes, S. Progress in Mathematics, vol. 152. Birkhäuser Boston, Inc., Boston, MA (1999)
Grove, K., Petersen, P., Wu, J.Y.: Geometric finiteness theorems via controlled topology. Invent. Math. 99(1):205–213 (1990), (Erratum in Invent. Math. 104(1):221–222 (1991))
Lang, U.: Local currents in metric spaces. To appear in J. Geom. Anal.
Petersen P.: A finiteness theorem for metric spaces. J. Differ. Geom. 31(2), 387–395 (1990)
Sormani, C., Wenger, S.: Weak convergence of currents and cancellation, with an appendix by Schul, R. and the second author, Calc. Var. Partial Differ. Equ. 38(1–2):183–206 (2010)
Wenger S.: Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2), 534–554 (2005)
Wenger, S.: Filling invariants at infinity and the Euclidean rank of Hadamard spaces, Int. Math. Res. Notices Volume 2006, Article ID 83090, 33 pp (2006)
Wenger S.: Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differ. Equ. 28(2), 139–160 (2007)
Wenger S.: Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math. 171(1), 227–255 (2008)
Wenger, S.: The asymptotic rank of metric spaces, to appear in Commentarii Mathematici Helvetici
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Wenger, S. Compactness for manifolds and integral currents with bounded diameter and volume. Calc. Var. 40, 423–448 (2011). https://doi.org/10.1007/s00526-010-0346-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-010-0346-y