Advertisement

Annals of Global Analysis and Geometry

, Volume 55, Issue 2, pp 197–213 | Cite as

Dual quadratic differentials and entire minimal graphs in Heisenberg space

  • José M. ManzanoEmail author
Article
  • 25 Downloads

Abstract

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces \({\mathbb {L}}(\kappa ,\tau )\) with isometry group of dimension 4, which are dual to the Abresch–Rosenberg differentials in the Riemannian counterparts \({\mathbb {E}}(\kappa ,\tau )\), and obtain some consequences. On the one hand, we classify explicitly those surfaces in \({\mathbb {L}}(\kappa ,\tau )\) with zero differential. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.

Keywords

Minimal surfaces Constant mean curvature Homogeneous 3-manifolds Heisenberg group Lorentz–Minkowski space Quadratic differentials Gauss curvature 

Mathematics Subject Classification

Primary 53A10 Secondary 53C30 

References

  1. 1.
    Abresch, U., Rosenberg, H.: A Hopf differential for constant mean curvature surfaces in \(\mathbf{S}^2\times \mathbf{R}\) and \(\mathbf{H}^2\times \mathbf{R}\). Acta Math. 193, 141–174 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abresch, U., Rosenberg, H.: Generalized Hopf differentials. Mat. Contemp. 28, 1–28 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Araújo, H., Leite, M.L.: How many maximal surfaces do correspond to one minimal surface? Math. Proc. Camb. Phil. Soc. 146(1), 165–175 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calabi, E.: Examples of Bernstein problems for some non-linear equations. In: Proceedings of Symposia in Pure Mathematics, American Mathematics Society, Providence, RI 15, 223–230 (1970)Google Scholar
  5. 5.
    Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. Math. (2) 104(3), 407–419 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choi, H.I., Treibergs, A.: Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differ. Geom. 32, 775–817 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daniel, B.: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82(1), 87–131 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Daniel, B.: The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Not. 2011(3), 674–695 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Daniel, B., Hauswirth, L., Mira, P.: Constant mean curvature surfaces in homogeneous manifolds Notes of the 4th KIAS Workshop on Differential Geometry, Seoul (2009)Google Scholar
  10. 10.
    Domínguez-Vázquez, M., Manzano, J.M.: Isoparametric surfaces in \({\mathbb{E}}(\kappa ,\tau )\)-spaces. Preprint arXiv:1706.09394
  11. 11.
    Espinar, J.M., Rosenberg, H.: Complete constant mean curvature surfaces in homogeneous spaces. Comment. Math. Helv. 86(3), 659–674 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fernández, I., Mira, P.: Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space. Trans. Am. Math. Soc. 361(11), 5737–5752 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Figueroa, C., Mercuri, F., Pedrosa, R.H.L.: Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl. (4) 177, 173–194 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hartman, P.: Parallel coordinates in the large. Am. J. Math. 86(4), 705–727 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, H.: Extensions of the duality between minimal surfaces and maximal surfaces. Geom. Dedicata 151, 373–386 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, H., Manzano, J. M.: Generalized Calabi’s correspondence and complete spacelike surfaces. Asian J. Math. (to appear)Google Scholar
  17. 17.
    Li, P.: Complete surfaces of at most quadratic area growth. Comment. Math. Helv. 72(1), 67–71 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    López, R.: Differential Geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom. 7(1), 44–107 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Manzano, J.M.: On the classification of Killing submersions and their isometries. Pac. J. Math. 270(2), 367–692 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Manzano, J.M. : On the conformal duality between constant mean curvature surfaces in \({\mathbb{E}}(\kappa ,\tau )\) and \({\mathbb{L}}(\kappa ,\tau )\). In: Proceedings of the Young Researcher Workshop on Differential Geometry in Minkowski Space, Universidad de Granada, pp. 89–118 (2017)Google Scholar
  21. 21.
    Manzano, J.M., Pérez, J., Rodríguez, M.: Parabolic stable surfaces with constant mean curvature. Calc. Var. Part. Differ. Equ. 42(1–2), 137–152 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Manzano, J.M., Nelli, B.: Height and area estimates for constant mean curvature graphs in \({\mathbb{E}}(\kappa,\tau )\)-spaces. J. Geom. Anal. 27(4), 3441–3473 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Meeks, W.H., Pérez, J., Ros, A.: Stable constant mean curvature surfaces. In: Ji, L., Li, P., Schoen, R., Simon, L. (eds.) Handbook of Geometrical Analysis, vol. 1, pp 301–380. International Press (2008) ISBN: 978-1-57146-130-8Google Scholar
  24. 24.
    Meeks, W. H., Mira, P., Pérez, J., Ros, A.: Constant mean curvature spheres in homogeneous three-manifolds. Preprint arXiv:1706.09394
  25. 25.
    O’Neill, B.: Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc., New York (1983) ISBN: 0-12-526740-1Google Scholar
  26. 26.
    Rosenberg, H., Souam, R., Toubiana, E.: General curvature estimates for stable \(H\)-surfaces in 3-manifolds and applications. J. Differ. Geom. 84(3), 623–648 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Torralbo, F.: Rotationally invariant constant mean curvature surfaces in homogeneous \(3\)-manifolds. Differ. Geom. Appl. 28(5), 593–607 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Treibergs, A.: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66(1), 39–56 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wan, T.Y.: Constant mean curvature surface harmonic map and universal Teichmuller space. J. Differ. Geom. 35, 643–657 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wan, T.Y., Au, T.K.: Parabolic constant mean curvature spacelike surfaces. Proc. Am. Math. Soc. 120, 559–564 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)MadridSpain

Personalised recommendations