Skip to main content
SpringerLink
Log in

The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

  • Original Article
  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Carathéodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group \({\mathbb{H}^n}\) . In particular we will positively answer to the Bernstein problem in the case n =  1 and we will provide counter examples when \({n\geq 5}\) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L.: Private Communication. March 2002

  2. Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. In: Calculus of Variations, non-smooth analysis and related topics. Set-Valued Anal. 10, 11–128 (2002)

  3. Ambrosio L. (2001). Some fine properties of sets of finite perimeter in Ahlfors regular metrice measure spaces. Adv. Math. 159: 51–67

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosio L., Serra Cassano F. and Vittone D. (2006). Intrinsic regular hypersurfaces in Heisenberg groups. J. Geom. Anal. 16(2): 187–232

    MATH  MathSciNet  Google Scholar 

  5. Balogh Z. (2003). Size of characteristic sets and functions with prescribed gradient. J. Reine Angew Math. 564: 63–83

    MATH  MathSciNet  Google Scholar 

  6. Balogh Z., Rickly M. and Serra Cassano F. (2003). Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric. Publ. Math. 47: 237–259

    MATH  MathSciNet  Google Scholar 

  7. Biroli M. and Mosco U. (1995). Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX.Ser. Rend. Lincei, Mat. Appl. 6: 37–44

    MATH  MathSciNet  Google Scholar 

  8. Buttazzo G., Giaquinta M. and Hildebrandt S. (1998). One-dimensional Variational Problems. An Introduction. Oxford University Press, New York

    MATH  Google Scholar 

  9. Capogna L., Danielli D. and Garofalo N. (1994). The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Comm. Anal. Geom. 12: 203–215

    MathSciNet  Google Scholar 

  10. Capogna, L., Danielli, D., Pauls, S.D., Tyson, J.T.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Preprint 2006

  11. Cheng J.H. and Hwang J.F. (2005). Properly embedded and immersed minimal surfaces in the Heisenberg group. Bull. Austr. Math. Soc. 70: 507–520

    Article  MathSciNet  Google Scholar 

  12. Cheng J.H., Hwang J.F., Malchiodi A. and Yang P. (2005). Minimal surfaces in in pseudohermitian geometry and the Bernstein problem in the Heisenberg group. Ann. Sc. Norm. Pisa Cl. Sci. 1: 129–177

    MathSciNet  Google Scholar 

  13. Cheng, J.H., Hwang, J.F., Yang, P.: Existence and uniqueness for p-area minimizers in the Heisenberg group. Preprint 2006

  14. Citti G. and Manfredini M. (2005). Blow-up in non homogeneous Lie groups and rectifiability. Houston J. Math. 31(2): 333–353

    MATH  MathSciNet  Google Scholar 

  15. Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. II. Interscience Publishers, New York (1962)

  16. Danielli, D., Garofalo, N., Nhieu, D.M.: Minimal Surfaces in Carnot groups. Preprint 2004

  17. Danielli, D., Garofalo, N., Nhieu, D.M.: A partial solution of the isoperimetric problem for the Heisenberg group. Preprint 2003

  18. Danielli, D., Garofalo, N., Nhieu, D.M.: A notable family of entire intrinsic minimal graphs in the Heinsenberg group which are not perimeter minimizing. Preprint 2005

  19. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, 1998

  20. Folland G.B. (1975). Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13: 161–207

    Article  MATH  MathSciNet  Google Scholar 

  21. Folland G.B. and Stein E.M. (1982). Hardy spaces on homogeneous groups. Princeton University Press, Princeton

    MATH  Google Scholar 

  22. Franchi B., Gallot S. and Wheeden R.L. (1994). Sobolev and isoperimetric inequalities for degenerate metrics. Math. Ann. 300: 557–571

    Article  MATH  MathSciNet  Google Scholar 

  23. Franchi B., Serapioni R. and Serra Cassano F. (1996). Meyers-Serrin Type Theorems and Relaxation of Variational Integrals Depending Vector Fields. Houston J. Math. 22(4): 859–889

    MATH  MathSciNet  Google Scholar 

  24. Franchi B., Serapioni R. and Serra Cassano F. (2001). Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321: 479–531

    Article  MATH  MathSciNet  Google Scholar 

  25. Franchi B., Serapioni R. and Serra Cassano F. (2003). On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. An. 13: 421–466

    MATH  MathSciNet  Google Scholar 

  26. Franchi, B., Serapioni, R., Serra Cassano, F.: Regular Submanifolds, Graphs and Area Formula in Heisenberg Groups. Adv. Math. (to appear)

  27. Garofalo N. and Nhieu D.M. (1996). Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49: 1081–1144

    Article  MATH  MathSciNet  Google Scholar 

  28. Garofalo, N., Pauls, S.: The Bernstein problem in the Heisenberg group. Preprint 2003

  29. Garofalo, N., Pauls, S.: Notable examples of H-minimal surfaces. Preprint 2005

  30. Garofalo, N., Selby, C.: Equations of mean curvature type in the Heisenberg group. work in progress 2005

  31. Giusti E. (1984). Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Boston

    MATH  Google Scholar 

  32. Gromov, M.: Carnot–Carathéodory spaces seen from within. In: Bellaiche, A., Risler, J. (eds) Subriemannian Geometry, Progress in Mathematics, Vol. 144. Birkhauser Verlag, Basel (1996)

  33. Hladky, R.K., Pauls, S.: Minimal surfaces in the roto-translation group with applications to a neurobiological image completion. Preprint 2005

  34. Hladky, R.K., Pauls, S.: Constant mean curvature surfaces in sub-Riemannian geometry. Preprint 2006

  35. Kirchheim, B., Serra Cassano, F.: Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 3, no. (4), 871–896 (2004)

  36. Magnani, V.: Elements of geometric Measure Theory on Sub-Riemannian Groups. Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 2002

  37. Mattila P. (2005). Measures with unique tangent measures in metric groups. Math. Scand. 97: 298–398

    MATH  MathSciNet  Google Scholar 

  38. Métivier G. (1992). Counterexamples to Hölgrem’s uniqueness for analytic non linear Cauchy problems. Inv. Math. 112: 217–222

    Article  Google Scholar 

  39. Montgomery, R.: A tour of Sub-Riemannian Geometries, their Geodesics and Applications. AMS, Providence (2002)

  40. Morgan F. (2000). Geometric Measure Theory. A Beginner’s Guide , 3rd edn. Academic, New York

    MATH  Google Scholar 

  41. Pansu P. (1989). Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129(1): 1–60

    Article  MathSciNet  Google Scholar 

  42. Pansu P. (1982). Une inégalité isopérimétrique sur le groupe de Heisenberg. C. R. Acad. Sci. Paris 295(2): 127–130

    MATH  MathSciNet  Google Scholar 

  43. Pauls S.D. (2004). A notion of rectifiability modelled on Carnot groups. Indiana Univ. Math. J. 53: 49–81

    Article  MATH  MathSciNet  Google Scholar 

  44. Pauls S.D. (2004). Minimal surfaces in the Heisenberg groups. Geom. Dedicata 104: 201–231

    Article  MATH  MathSciNet  Google Scholar 

  45. Pauls, S.D.: H-minimal graphs of low regularity in \({\mathbb{H}^1}\) . Comm. Math. Helv. 81, 337–381 (2006)

    Google Scholar 

  46. Ritoré, M., Rosales, C.: Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg groups \({\mathbb{H}^n}\) . Preprint 2005

  47. Ritoré, M., Rosales, C.: Area-stationary surfaces in the Heisenberg group \({\mathbb{H}^1}\) . Preprint 2005

  48. Stein E.M. (1993). Harmonic Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050, Povo (Trento), Italy

    Vittorio Barone Adesi & Francesco Serra Cassano

  2. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Davide Vittone

Authors
  1. Vittorio Barone Adesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesco Serra Cassano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Davide Vittone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davide Vittone.

Additional information

V.B.A. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy.

F.S.C. is supported by MIUR, Italy, GNAMPA of INDAM and University of Trento, Italy.

D.V. is supported by MIUR, Italy, GNAMPA of INDAM and Scuola Normale Superiore, Italy. Part of the work was done while D.V. was a visitor at the University of Trento. He wishes to thank the Department of Mathematics for its hospitality.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adesi, V.B., Cassano, F.S. & Vittone, D. The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations. Calc. Var. 30, 17–49 (2007). https://doi.org/10.1007/s00526-006-0076-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-006-0076-3

Keywords

Search

Navigation