Inventiones mathematicae

, Volume 196, Issue 3, pp 733–771 | Cite as

Null curves and directed immersions of open Riemann surfaces


Mathematics Subject Classification

32E10 32E30 32H02 32Q28 14H50 14Q05 49Q05 



A. Alarcón is supported by Vicerrectorado de Política Científica e Investigación de la Universidad de Granada, and is partially supported by MCYT-FEDER grants MTM2007-61775 and MTM2011-22547, Junta de Andalucía Grant P09-FQM-5088, and the grant PYR-2012-3 CEI BioTIC GENIL (CEB09-0010) of the MICINN CEI Program.

F. Forstnerič is supported by the research program P1-0291 from ARRS, Republic of Slovenia.

We wish to thank the anonymous referee for the useful remarks which helped us to improve the presentation.


  1. 1.
    Abraham, R.: Transversality in manifolds of mappings. Bull. Am. Math. Soc. 69, 470–474 (1963) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ahlfors, L.V., Sario, L.: Riemann Surfaces. Princeton Mathematical Series, vol. 26. Princeton University Press, Princeton (1960) MATHGoogle Scholar
  3. 3.
    Alarcón, A.: Compact complete minimal immersions in \(\mathbb {R}^{3}\). Trans. Am. Math. Soc. 362, 4063–7076 (2010) CrossRefMATHGoogle Scholar
  4. 4.
    Alarcón, A., Ferrer, L., Martín, F.: Density theorems for complete minimal surfaces in \(\mathbb {R}^{3}\). Geom. Funct. Anal. 18, 1–49 (2008) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Alarcón, A., Forstnerič, F.: Every bordered Riemann surface is a complete proper curve in a ball. Math. Ann. (2013).
  6. 6.
    Alarcón, A., López, F.J.: Null curves in \(\mathbb{C}^{3}\) and Calabi-Yau conjectures. Math. Ann. 355, 429–455 (2013) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Alarcón, A., López, F.J.: Minimal surfaces in \(\mathbb{R}^{3}\) properly projecting into \(\mathbb{R}^{2}\). J. Differ. Geom. 90, 351–382 (2012) MATHGoogle Scholar
  8. 8.
    Alarcón, A., López, F.J.: Compact complete null curves in complex 3-space. Israel J. Math. (2013)., arXiv:1106.0684
  9. 9.
    Bryant, R.: Surfaces of mean curvature one in hyperbolic space. In: Théorie des Variétés Minimales et Applications (Palaiseau, 1983–1984). Astérisque 154155 (1987), 12, 321–347, 353 (1988) Google Scholar
  10. 10.
    Chirka, E.M.: Complex Analytic Sets. Kluwer, Dordrecht (1989) CrossRefMATHGoogle Scholar
  11. 11.
    Colding, T.H., Minicozzi, W.P. II: The Calabi-Yau conjectures for embedded surfaces. Ann. Math. (2) 167, 211–243 (2008) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Collin, P., Hauswirth, L., Rosenberg, H.: The geometry of finite topology Bryant surfaces. Ann. Math. (2) 153, 623–659 (2001) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Drinovec Drnovšek, B., Forstnerič, F.: Holomorphic curves in complex spaces. Duke Math. J. 139, 203–254 (2007) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Eliashberg, Y., Mishachev, N.: Introduction to the h-Principle. Graduate Studies in Math., vol. 48. Amer. Math. Soc., Providence (2002) Google Scholar
  15. 15.
    Ferrer, L., Martín, F., Meeks, W.H. III: Existence of proper minimal surfaces of arbitrary topological type. Adv. Math. 231, 378–413 (2012) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81. Springer, New York (1991) Google Scholar
  17. 17.
    Forstnerič, F.: The Oka principle for sections of subelliptic submersions. Math. Z. 241, 527–551 (2002) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Forstnerič, F.: The Oka principle for multivalued sections of ramified mappings. Forum Math. 15, 309–328 (2003) MATHMathSciNetGoogle Scholar
  19. 19.
    Forstnerič, F.: Manifolds of holomorphic mappings from strongly pseudoconvex domains. Asian J. Math. 11, 113–126 (2007) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Forstnerič, F.: Oka manifolds. C. R. Acad. Sci. Paris, Ser. I 347, 1017–1020 (2009) CrossRefMATHGoogle Scholar
  21. 21.
    Forstnerič, F.: Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis). Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, vol. 56. Springer, Berlin (2011) CrossRefMATHGoogle Scholar
  22. 22.
    Forstnerič, F., Lárusson, F.: Survey of Oka theory. N.Y. J. Math. 17a, 1–28 (2011) Google Scholar
  23. 23.
    Forstnerič, F.: Oka manifolds: From Oka to Stein and back. With an appendix by F. Lárusson. J. Math. Fac. Sci. Toulouse, in press. arxiv:1211.6383
  24. 24.
    Forstnerič, F., Lárusson, F.: Holomorphic flexibility properties of compact complex surfaces. Int. Math. Res. Not. (2013). doi: 10.1093/imrn/rnt044 Google Scholar
  25. 25.
    Grauert, H.: Approximationssätze für holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. Math. Ann. 133, 450–472 (1957) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Grauert, H.: Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen. Math. Ann. 133, 139–159 (1957) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Grauert, H.: Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann. 135, 263–273 (1958) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Gromov, M.: Convex integration of differential relations, I. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 329–343 (1973) (Russian). English transl.: Math. USSR Izv. 37 (1973) MATHMathSciNetGoogle Scholar
  29. 29.
    Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 9. Springer, Berlin (1986) CrossRefMATHGoogle Scholar
  30. 30.
    Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2, 851–897 (1989) MATHMathSciNetGoogle Scholar
  31. 31.
    Gunning, R.C., Narasimhan, R.: Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North Holland, Amsterdam (1990) MATHGoogle Scholar
  33. 33.
    Lempert, L.: The dolbeault complex in infinite dimensions. I. J. Am. Math. Soc. 11, 485–520 (1998) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    López, F.J., Martín, F., Morales, S.: Adding handles to Nadirashvili’s surfaces. J. Differ. Geom. 60, 155–175 (2002) MATHGoogle Scholar
  35. 35.
    López, F.J., Ros, A.: On embedded complete minimal surfaces of genus zero. J. Differ. Geom. 33, 293–300 (1991) MATHGoogle Scholar
  36. 36.
    Majcen, I.: Closed holomorphic 1-forms without zeros on Stein manifolds. Math. Z. 257, 925–937 (2007) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Martín, F., Umehara, M., Yamada, K.: Complete bounded null curves immersed in \(\mathbb {C}^{3}\) and \(\mathit{SL}(2,\mathbb {C})\). Calc. Var. Partial Differ. Equ. 36, 119–139 (2009). Erratum: Complete bounded null curves immersed in \(\mathbb {C}^{3}\) and \(\mathit{SL}(2,\mathbb {C})\). Calc. Var. Partial Differential Equations 46, 439–440 (2013) CrossRefMATHGoogle Scholar
  38. 38.
    Meeks, W.H. III, Pérez, J.: The classical theory of minimal surfaces. Bull. Am. Math. Soc. (N.S.) 48, 325–407 (2011) CrossRefMATHGoogle Scholar
  39. 39.
    Meeks, W.H. III, Pérez, J.: A Survey on Classical Minimal Surface Theory. University Lecture Series, vol. 60. Amer. Math. Soc., Providence (2012) MATHGoogle Scholar
  40. 40.
    Meeks, W.H. III, Pérez, J., Ros, A.: The embedded Calabi-Yau conjectures for finite genus.
  41. 41.
    Nadirashvili, N.: Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Invent. Math. 126, 457–465 (1996) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Oka, K.: Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin. J. Sci. Hiroshima Univ. 9, 7–19 (1939) MATHMathSciNetGoogle Scholar
  43. 43.
    Osserman, R.: A Survey of Minimal Surfaces, 2nd edn. Dover, New York (1986) Google Scholar
  44. 44.
    Rosenberg, H.: Bryant surfaces. In: The global theory of minimal surfaces in flat spaces. Lectures given at the 2nd C.I.M.E. Session held in Martina Franca, July 7—14, 1999. Lecture Notes in Math., vol. 1775, pp. 67—111. Springer-Verlag, Berlin (2002) Google Scholar
  45. 45.
    Umehara, M., Yamada, K.: Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space. Ann. Math. (2) 137, 611–638 (1993) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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