Reifenberg’s isoperimetric inequality revisited



We prove a generalization of Reifenberg’s isoperimetric inequality. The main result of this paper is used in Harrison and Pugh (General methods of elliptic minimization, Available on arxiv, 2016) to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.

Mathematics Subject Classification




  1. 1.
    Reifenberg, E.R.: Solution of the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hurewicz, W., Wallman, H.: Dimension Theory. Princeton University Press, Princeton (1948)zbMATHGoogle Scholar
  3. 3.
    Harrison, J., Pugh, H.: Spanning via Cech cohomology. arXiv eprints (2014)Google Scholar
  4. 4.
    Frederick, J.: Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. 87(2), 321–391 (1968)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Eilenberg, S.: On \({\varphi }\) measures. Ann. Soc. Pol. de Math 17, 251–252 (1938)Google Scholar
  6. 6.
    De Lellis, C., Ghiraldin, F., Maggi, F.: A direct approach to Plateau’s problem. J. Eur. Math. Soc. 288, 1–17 (2015)zbMATHGoogle Scholar
  7. 7.
    Sharpe, R.W.: Differential Geometry: Cartan’s generalization of Klein’s Erlangen program. Springer, New York (1997)zbMATHGoogle Scholar
  8. 8.
    Harrison, J., Pugh, H.: General methods of elliptic minimization. Available on arxiv (2016)Google Scholar
  9. 9.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA

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