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The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg group

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Abstract

We prove that, in the first Heisenberg group \(\mathbb {H}\), an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of \(\mathbb {H}\). Moreover two examples are given for stressing result’s sharpness.

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References

  1. Alberti, G., Bianchini, S., Caravenna, L.: Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I. J. Differ. Equ. 261(8), 4298–4337 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159(1), 51–67 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Ghezzi, R.: Sobolev and bounded variation functions on metric measure spaces. In: Geometry, Analysis and Dynamics on sub-Riemannian manifolds. Vol. II. EMS Series Lecture Mathematics Europe Mathematics Society, Zurich, pp. 211–273 (2016)

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and its Applications, vol. 25, pp. 8–133. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  6. Adesi, V.B., Serra Cassano, F., Vittone, D.: The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations. Cal. Var. Part. Differ. Equ. 30(1), 17–49 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bigolin, F., Caravenna, L., Serra Cassano, F.: Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation. Ann. Inst. H. Poincare Anal. Non Lineaire 32(5), 925–963 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Capogna, L., Citti, G., Manfredini, M.: Regularity of non-characteristic minimal graphs in the Heisenberg group H1. Indiana Univ. Math. J. 58(5), 2115–2160 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Capogna, L., Citti, G., Manfredini, M.: Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups Hn, n \(>\) 1. J. Reine Angew. Math. 648, 75–110 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Capogna, L., Danielli, D., Garofalo, N.: The geometric Sobolev embedding for vector fields and the isoperimetric inequality. Comm. Anal. Geom. 2(2), 203–215 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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, pp. 14–223. Birkhauser, Basel (2007)

    MATH  Google Scholar 

  12. Cheng, J.-H., Hwang, J.-F., Yang, P.: Existence and uniqueness for p-area minimizers in the Heisenberg group. Math. Ann. 337(2), 253–293 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, J.-H., Hwang, J.-F., Yang, P.: Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group. Math. Ann. 344(1), 1–35 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cheng, J.-H., Hwang, J.-F., Malchiodi, A., Yang, P.: Minimal surfaces in pseudohermitian geometry. Ann. Sci. Norm. Super. Pisa Cl. Sci. 4(1), 129–177 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Danielli, D., Garofalo, N., Nhieu, D.M., Pauls, S.D.: Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1. J. Differ. Geom. 81(2), 251295 (2009)

    Article  MATH  Google Scholar 

  16. Danielli, D., Garofalo, N., Nhieu, D.-M., Pauls, S.D.: The Bernstein problem for embedded surfaces in the Heisenberg group H1. Indiana Univ. Math. J. 59(2), 563–594 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, p. 14. CRC Press, Boca Raton (2015)

    Book  MATH  Google Scholar 

  18. Fogagnolo, M., Monti, R., Vittone, D.: Variation formulas for H-rectifiable sets. Ann. Acad. Sci. Fenn. Math. 42.1, 239256 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321.3, 479531 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Galli, M.: The regularity of Euclidean Lipschitz boundaries with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds. Nonlinear Anal. 136, 40–50 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Galli, M., Ritoré, M.: Area-stationary and stable surfaces of class C1 in the sub-Riemannian Heisenberg group H1. Adv. Math. 285, 737–765 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Galli, M., Ritoré, M.: Regularity of C1 surfaces with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds. Calc. Var. Part. Differ. Equ. 54.3, 25032516 (2015)

    MATH  Google Scholar 

  23. Gariepy, R., Ziemer, W.: Modern Real Analysis. Springer, Berlin (1995)

    Google Scholar 

  24. Garofalo, N., Pauls, S. D.: The Bernstein Problem in the Heisenberg Group. In: ArXiv Mathematics e-prints (2002). eprint: math/ 0209065

  25. Garofalo, N., Nhieu, D.-M.: Isoperimetric and Sobolev inequalities for Carnot–Caratheodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49(10), 1081–1144 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Golo, S.: Some remarks on contact variations in the first Heisenberg group. Ann. Acad. Sci. Fenn. Math. 43(1), 311–335 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hartman, P.: Ordinary Differential Equations. Classics in Applied Mathematics, vol. 38, p. 20. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

    Book  MATH  Google Scholar 

  28. Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, vol. 2096. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  29. Hurtado, A., Ritoré, M., Rosales, C.: The classification of complete stable area-stationary surfaces in the Heisenberg group H1. Adv. Math. 224(2), 561–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kirchheim, B., Serra Cassano, F.: Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3(4), 871–896 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Monti, R.: Isoperimetric problem and minimal surfaces in the Heisenberg group. In: Geometric Measure Theory and Real Analysis. CRM Series, vol. 17, pp. 57–129. Edizioni della Normale, Pisa (2014)

    MATH  Google Scholar 

  32. Monti, R.: Minimal surfaces and harmonic functions in the Heisenberg group. Nonlinear Anal. 126, 378–393 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Monti, R., Serra Cassano, F., Vittone, D.: A negative answer to the Bernstein problem for intrinsic graphs in the Heisenberg group Boll. Unione Mat. Ital. 1(3), 709–727 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Monti, R., Stefani, G.: Improved Lipschitz approximation of H-perimeter minimizing boundaries. J. Math. Pures Appl. 108.3, 372–398 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Monti, R., Vittone, D.: Height estimate and slicing formulas in the Heisenberg group. Anal. PDE 8(6), 1421–1454 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pansu, P.: Geometrie du groupe de Heisenberg. PhD thesis. Universite Paris 7 (1982)

  37. Pansu, P.: Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un. Ann. of Math. 129.1, 1–60 (1989)

    Article  MATH  Google Scholar 

  38. Pansu, P.: Une inegalite isoperimetrique sur le groupe de Heisenberg. C. R. Acad. Sci. Paris Ser. I Math. 295.2, 127–130 (1982)

    MATH  Google Scholar 

  39. Pauls, S.D.: H-minimal graphs of low regularity in H1. Comment. Math. Helv. 81(2), 337–381 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pauls, S.D.: Minimal surfaces in the Heisenberg group. Geom. Dedicata 104, 201–231 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ritoré, M., Rosales, C.: Area-stationary and stable surfaces in the sub-Riemannian Heisenberg group H1. Mat. Contemp. 35, 185–203 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Ritoré, M.: Examples of area-minimizing surfaces in the sub-Rieman-nian Heisenberg group H1 with low regularity. Calc. Var. Part. Differ. Equ. 34(2), 179–192 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Serra Cassano, F.: Some topics of geometric measure theory in Carnot groups. In: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. EMS Series of Lectures in Mathematics European Mathematical Society, Zurich (2016)

  44. Serra Cassano, F., Vittone, D.: Graphs of bounded variation, existence and local boundedness of non-parametric minimal surfaces in Heisenberg groups. Adv. Calc. Var. 7(4), 409–492 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ziemer, W.P.: Modern real analysis Graduate Texts in Mathematics. Graduate Texts in Mathematics. With contributions by Monica Torres, vol. 278, 2nd edn, p. 9. Springer, Cham (2017)

    Google Scholar 

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Acknowledgements

This paper benefited from fruitful discussions with M. Ritoré: The authors want to thank him. The authors want to thank also the anonymous referee for the careful reading and suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    Sebastiano Nicolussi

  2. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123, Trento, Italy

    Francesco Serra Cassano

Authors
  1. Sebastiano Nicolussi
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  2. Francesco Serra Cassano
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Correspondence to Sebastiano Nicolussi.

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Communicated by L. Ambrosio.

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Both authors have been supported by the European Unions Seventh Framework Programme, Marie Curie Actions-Initial Training Network, under grant Agreement No. 607643, “Metric Analysis For Emergent Technologies (MAnET)”.

S. N. has been supported by EPSRC Grant “Sub-Elliptic Harmonic Analysis” (EP/P002447/1), and by University of Padova STARS Project “Sub-Riemannian Geometry and Geometric Measure Theory Issues: Old and New”.

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

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Nicolussi, S., Serra Cassano, F. The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg group. Calc. Var. 58, 141 (2019). https://doi.org/10.1007/s00526-019-1581-5

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