Journal of Geometric Analysis

, Volume 23, Issue 2, pp 490–538 | Cite as

Flat Forms in Banach Spaces

  • Marie A. SnipesEmail author


We define a flat partial differential form in a Banach space and show that the space of these forms is isometrically the dual space of the space of flat chains as defined by T. Adams.


Flat chain Flat cochain Flat form Partial form Banach space 

Mathematics Subject Classification (2000)2010

28A75 28C05 49Q15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, T.: Flat chains in Banach spaces. J. Geom. Anal. 18(1), 1–28 (2008) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Álvarez Paiva, J.C., Thompson, A.C.: Volumes on normed and Finsler spaces. In: A Sampler of Riemann-Finsler Geometry, Math. Sci. Res. Inst. Publ., vol. 50, pp. 1–48. Cambridge Univ. Press, Cambridge (2004) Google Scholar
  3. 3.
    Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry. Math. Sci. Res. Inst. Publ., vol. 31, pp. 1–58. Cambridge Univ. Press, Cambridge (1997) Google Scholar
  5. 5.
    Cartan, H.: Differential Forms. Houghton Mifflin, Boston (1970) zbMATHGoogle Scholar
  6. 6.
    Cartan, H.: Differential Calculus. Hermann, Paris (1971) zbMATHGoogle Scholar
  7. 7.
    De Pauw, T., Hardt, R.: Rectifiable and flat G-chains in metric spaces. Am. J. Math. (to appear) Google Scholar
  8. 8.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969) zbMATHGoogle Scholar
  9. 9.
    Greub, W.: Multilinear Algebra, 2nd edn. Springer, New York (1978) zbMATHCrossRefGoogle Scholar
  10. 10.
    Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Heinonen, J.: Lectures on Lipschitz Analysis. Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100. University of Jyväskylä, Jyväskylä (2005) zbMATHGoogle Scholar
  12. 12.
    Heinonen, J., Keith, S.: Flat forms, bi-Lipschitz parametrizations and smoothability of manifolds. Work in preparation Google Scholar
  13. 13.
    Heinonen, J., Pankka, P., Rajala, K.: Quasiconformal frames. Preprint 359, Department of Mathematics and Statistics, Jyväskylä (2007) Google Scholar
  14. 14.
    Heinonen, J., Sullivan, D.: On the locally branched Euclidean metric gauge. Duke Math. J. 114(1), 15–41 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lang, U.: Local Currents in Metric Spaces. Lecture Notes from the 17th Jyväskylä Summer School Google Scholar
  16. 16.
    Lindenstrauss, J., Tzfrari, L.: Classical Banach Spaces I and II. Springer, New York (1996) Google Scholar
  17. 17.
    Noltie, S.V.: Chains in Banach spaces. In: Real and Stochastic Analysis. Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., pp. 211–248. Wiley, New York (1986) Google Scholar
  18. 18.
    Royden, H.L.: Real Analysis, 3nd edn. Macmillan, New York (1988) zbMATHGoogle Scholar
  19. 19.
    Wenger, S.: Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2), 534–554 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Wenger, S.: Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differ. Equ. 28(2), 139–160 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957) zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2011

Authors and Affiliations

  1. 1.Department of MathematicsKenyon CollegeGambierUSA

Personalised recommendations