Archive for Rational Mechanics and Analysis

, Volume 223, Issue 3, pp 1123–1182 | Cite as

Area Minimizing Discs in Metric Spaces

  • Alexander Lytchak
  • Stefan Wenger


We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acerbi E., Fusco N.: Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86((2), 125–145 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Allcock D.: An isoperimetric inequality for the Heisenberg groups. Geom. Funct. Anal. 8(2), 219–233 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Almgren, Jr., F.J.: Almgren’s Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet’S Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2. World Scientific Monograph Series in Mathematics, Vol. 1. World Scientific Publishing Co., Inc., River Edge, 2000. (With a preface by Jean E. Taylor and Vladimir Scheffer)Google Scholar
  4. 4.
    Alvarez Paiva, J.C., Thompson, A.C.: Volumes on normed and Finsler spaces. A sampler of Riemann–Finsler geometry, Vol. 50 of Math. Sci. Res. Inst. Publ., pp. 1–48. Cambridge Univ. Press, Cambridge, 2004Google Scholar
  5. 5.
    Ambrosio L.: Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17(3), 439–478 (1990)MathSciNetMATHGoogle Scholar
  6. 6.
    Ambrosio, L., De Lellis, C., Schmidt, T.: Partial regularity for mass-minimizing currents in hilbert spaces. J. Reine Angew. Math. (To appear)Google Scholar
  7. 7.
    Ambrosio L., Kirchheim B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ambrosio L., Kirchheim B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ambrosio L., Schmidt T.: Compactness results for normal currents and the Plateau problem in dual Banach spaces. Proc. Lond. Math. Soc. (3) 106(5), 1121–1142 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ambrosio, L., Tilli, P. Topics on Analysis in Metric Spaces, Vol. 25 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004Google Scholar
  11. 11.
    Bacher K., Sturm K.-T.: Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259(1), 28–56 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ball, K.: An elementary introduction to modern convex geometry. Flavors of Geometry, Vol. 31 of Math. Sci. Res. Inst. Publ., pp. 1–58. Cambridge Univ. Press, Cambridge, 1997Google Scholar
  13. 13.
    Balogh, Z. M., Monti, R., Tyson, J. T.: Frequency of Sobolev and quasiconformal dimension distortion. J. Math. Pures Appl. (9) 99(2), 125–149 (2013)Google Scholar
  14. 14.
    Bernig A.: Centroid bodies and the convexity of area functionals. J. Differ. Geom. 98(3), 357–373 (2014)MathSciNetMATHGoogle Scholar
  15. 15.
    Bridson, M. R., Haefliger, A.: Metric Spaces of Non-Positive Curvature, Vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1999Google Scholar
  16. 16.
    Burago, D., Ivanov, S.: On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume. Ann. of Math. (2). 156(3), 891–914 (2002)Google Scholar
  17. 17.
    Burago D., Ivanov S.: Minimality of planes in normed spaces. Geom. Funct. Anal. 22(3), 627–638 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    David, G.: Should we solve plateau’s problem again? Advances in Analysis. The Legacy of Elias M. Stein. Ed. Fefferman, Charles and Ionescu, A.D. and Phong, D.H. and Wainger, S, Vol. 50 of Princeton Mathematical Series, pp. 108–145. Princeton University Press, Princeton 2014Google Scholar
  19. 19.
    Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal surfaces, Second ed., Vol. 339 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2010. (With assistance and contributions by A. Küster and R. Jakob)Google Scholar
  20. 20.
    Douglas J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33(1), 263–321 (1931)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, Vol. 1. North-Holland Publishing Co., Amsterdam, 1976. (Translated from the French, Studies in Mathematics and its Applications)Google Scholar
  22. 22.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992Google Scholar
  23. 23.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York Inc., New York, 1969Google Scholar
  24. 24.
    Federer H., Fleming W.H.: Normal and integral currents. Ann. of Math. (2) 72, 458–520 (1960)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gigli, N.: The splitting theorm in non-smooth context. Preprint arXiv:1302.5555
  26. 26.
    Gigli, N., Mondino, A., Rajala, T.: Euclidean spaces as weak tangents of infinitesimally hilbertian metric measure spaces with ricci curvature bounded below. J. Reine Angew. Math. (To appear).Google Scholar
  27. 27.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin, 2001. (Reprint of the 1998 edition)Google Scholar
  28. 28.
    Gromov M.: Filling Riemannian manifolds. J. Differential Geom. 18(1), 1–147 (1983)MathSciNetMATHGoogle Scholar
  29. 29.
    Hajłasz P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)MathSciNetMATHGoogle Scholar
  30. 30.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev Spaces on Metric Measure Spaces, Vol. 27 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2015Google Scholar
  31. 31.
    Heinonen J., Koskela P., Shanmugalingam N., Tyson J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Isbell J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39, 65–76 (1964)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ivanov S.V.: Volumes and areas of Lipschitz metrics. Algebra i Analiz. 20(3), 74–111 (2008)MathSciNetGoogle Scholar
  34. 34.
    Karmanova M.B.: Area and co-area formulas for mappings of the Sobolev classes with values in a metric space. Sibirsk. Mat. Zh. 48(4), 778–788 (2007)MathSciNetMATHGoogle Scholar
  35. 35.
    Kinnunen J.: Higher integrability with weights. Ann. Acad. Sci. Fenn. Ser. A I Math. 19(2), 355–366 (1994)MathSciNetMATHGoogle Scholar
  36. 36.
    Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1), 113–123 (1994)Google Scholar
  37. 37.
    Korevaar N.J., Schoen R.M.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1(3–4), 561–659 (1993)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lang U.: Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5(3), 297–331 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Logaritsch, P., Spadaro, E.: A representation formula for the p-energy of metric space-valued Sobolev maps. Commun. Contemp. Math. 14(6), 1250043, 10 (2012)Google Scholar
  40. 40.
    Lytchak A.: Differentiation in metric spaces. Algebra i Analiz. 16(6), 128–161 (2004)MathSciNetGoogle Scholar
  41. 41.
    Lytchak, A., Wenger, S.: Isoperimetric characterization of upper curvature bounds. (In preparation)Google Scholar
  42. 42.
    Lytchak, A., Wenger, S.: Energy and area minimizers in metric spaces. Adv. Calc. Var. (2016). Online first, doi: 10.1515/acv-2015-0027
  43. 43.
    Lytchak, A., Wenger, S.: Intrinsic structure of minimal discs in metric spaces. (2016). Preprint arXiv:1602.06755v1
  44. 44.
    Mese C., Zulkowski P.R.: The Plateau problem in Alexandrov spaces. J. Differ. Geom. 85(2), 315–356 (2010)MathSciNetMATHGoogle Scholar
  45. 45.
    Morgan F., Ritoré M.: Isoperimetric regions in cones. Trans. Am. Math. Soc. 354(6), 2327–2339 (2002)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Morrey, Jr., C.B.: The problem of Plateau on a Riemannian manifold. Ann. Math. (2). 49, 807–851 (1948)Google Scholar
  47. 47.
    Nikolaev, I.G.: Solution of the Plateau problem in spaces of curvature at most K. Sibirsk. Mat. Zh. 20(2), 345–353, 459 (1979)Google Scholar
  48. 48.
    Overath P., von der Mosel H.: Plateau’s problem in Finsler 3-space. Manuscripta Math. 143(3–4), 273–316 (2014)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Perel’man, G.Y., Petrunin, A.M.: Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem. Algebra i Analiz. 5(1), 242–256 (1993)Google Scholar
  50. 50.
    Radó T.: On Plateau’s problem. Ann. Math. (2) 31((3), 457–469 (1930)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. Sibirsk. Mat. Zh. 38(3), 657–675, iii–iv (1997)Google Scholar
  52. 52.
    Reshetnyak Y.G.: Sobolev classes of functions with values in a metric space. II. Sibirsk. Mat. Zh. 45(4), 855–870 (2004)MATHGoogle Scholar
  53. 53.
    Reshetnyak Y.G.: On the theory of Sobolev classes of functions with values in a metric space. Sibirsk. Mat. Zh. 47(1), 46–168 (2006)MathSciNetMATHGoogle Scholar
  54. 54.
    Tukia, P.: The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A I Math. 51, 49–72 (1980)Google Scholar
  55. 55.
    Wenger S.: Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2), 534–554 (2005)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Wenger S.: Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differ. Equ. (2, 139–160 (2007)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Wenger S.: Characterizations of metric trees and Gromov hyperbolic spaces. Math. Res. Lett. 15(5), 1017–1026 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Wenger S.: Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math. 171(1), 227–255 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Wenger, S.: Plateau’s problem for integral currents in locally non-compact metric spaces. Adv. Calc. Var. 7(2), 227–240 (2014)Google Scholar
  60. 60.
    Ziemer, W. P.: Weakly Differentiable Functions, Vol. 120 of Graduate Texts in Mathematics. Springer, New York, 1989. (Sobolev spaces and functions of bounded variation)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität KölnKölnGermany
  2. 2.Department of MathematicsUniversity of FribourgFribourgSwitzerland

Personalised recommendations