A proof by calibration of an isoperimetric inequality in the Heisenberg group \({\mathbb{H}^n}\)

Article

Abstract

Let D be a closed disk centered at the origin in the horizontal hyperplane {t = 0} of the sub-Riemannian Heisenberg group \({\mathbb{H}^n}\), and C the vertical cylinder over D. We prove that the perimeter of any set E such that \({D\subset E\subset C}\) is larger than or equal to the one of the rotationally symmetric sphere with constant mean curvature of the same volume, and that equality holds only for these spheres using a recent result by Monti and Vittone (Math Z 1–17, 2010).

Mathematics Subject Classification (2000)

53C17 53C42 49Q20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosio L., Serra Cassano F., Vittone D.: Intrinsic regular hypersurfaces in Heisenberg groups. J. Geom. Anal. 16(2), 187–232 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Capogna L., Danielli D., Pauls S.D., Tyson J.T.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, vol. 259. Birkhäuser Verlag, Basel (2007)Google Scholar
  3. 3.
    Danielli D., Garofalo N., Nhieu D.-M.: A partial solution of the isoperimetric problem for the Heisenberg group. Forum Math. 20(1), 99–143 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Franchi B., Serapioni R., Serra Cassano F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321(3), 479–531 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Giusti E.: Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)Google Scholar
  6. 6.
    Leonardi G.P., Masnou S.: On the isoperimetric problem in the Heisenberg group \({\mathbb H^n}\). Ann. Mat. Pura Appl. (4) 184(4), 533–553 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Leonardi G.P., Rigot S.: Isoperimetric sets on Carnot groups. Houst. J. Math. 29(3), 609–637 (2003) (electronic)MathSciNetMATHGoogle Scholar
  8. 8.
    Monti, R.: Some properties of Carnot–Carathéodory balls in the Heisenberg group, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 3, 155–167 (2001)Google Scholar
  9. 9.
    Monti R.: Brunn–Minkowski and isoperimetric inequality in the Heisenberg group. Ann. Acad. Sci. Fenn. Math. 28(1), 99–109 (2008)MathSciNetGoogle Scholar
  10. 10.
    Monti R.: Heisenberg isoperimetric problem. The axial case. Adv. Calc. Var. 1, 93–121 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Monti R., Rickly M.: Convex isoperimetric sets in the Heisenberg group. Ann. Sci. Norm. Super. Pisa Cl. Sci. 5 8(2), 391–415 (2009)MathSciNetMATHGoogle Scholar
  12. 12.
    Monti, R., Vittone, D.: Sets with finite \({\mathbb{H}}\) -perimeter and controlled normal, Math. Z. 1–17 (2010). doi: 10.1007/s00209-010-0801-7
  13. 13.
    Ni Y.: Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits. Ann. Mat. Pura Appl. 183(4), 555–570 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Pansu, P.: An isoperimetric inequality on the Heisenberg group. Rend. Sem. Mat. Univ. Politec. Torino (1983), no. Special Issue, 159–174 (1984). Conference on Differential Geometry on Homogeneous Spaces (Turin, 1983)Google Scholar
  15. 15.
    Ritoré M.: Area-stationary surfaces in the Heisenberg group \({\mathbb H^1}\). Adv. Math. 219(2), 633–671 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ritoré, M.: Mean curvature flow, isoperimetric inequalities and hyperbolic geometry, In: Miquel, V., Portí, J. (eds.) Mean Curvature Flow and Isoperimetric Inequalities, Advanced Courses in Mathematics—CRM Barcelona. Birkhäuser, Basel (2010)Google Scholar
  17. 17.
    Ritoré M., Rosales C.: Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group \({\mathbb H^n}\). J. Geom. Anal. 16(4), 703–720 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Schmidt E.: Über das isoperimetrische Problem im Raum von n Dimensionen. Math. Z. 44, 689–788 (1939)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Schmidt E.: Die isoperimetrischen Ungleichungen auf der gewöhnlichen Kugel und für Rotationskörper im n-dimensionalen sphärischen Raum. Math. Z. 46, 743–794 (1940)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schmidt E.: Über die isoperimetrische Aufgabe im n-dimensionalen Raum konstanter negativer Kuümmung. I. Die isoperimetrischen Ungleichungen in der hyperbolischen Ebene und für Rotationskörper im n-dimensionalen hyperbolischen Raum. Math. Z. 46, 204–230 (1940)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schmidt E.: Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl. Math. Z. 49, 1–109 (1943)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Schwarz H.A.: Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumens. Nachrichten von der Königlichen Gesellschaft der Wissenchaften und der Georg-Augusts-Universität zu Göttingen 1, 1–13 (1884)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaEspaña

Personalised recommendations