Harmonic mappings and conformal minimal immersions of Riemann surfaces into \({\mathbb {R}^{\rm N}}\)

  • Antonio Alarcón
  • Isabel Fernández
  • Francisco J. López


We prove that for any open Riemann surface \({\mathcal{N}}\), natural number N ≥ 3, non-constant harmonic map \({h:\mathcal{N} \to \mathbb{R}}\) N−2 and holomorphic 2-form \({\mathfrak{H}}\) on \({\mathcal{N}}\) , there exists a weakly complete harmonic map \({X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) with Hopf differential \({\mathfrak{H}}\) and \({(X_j)_{j=3,\ldots,{\sc N}}=h.}\) In particular, there exists a complete conformal minimal immersion \({Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}\) such that \({(Y_j)_{j=3,\ldots,{\sc N}}=h}\) . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of \({\mathbb{CP}^{{\sc N}-1}}\) in general position. (2) There exist complete non-proper embedded minimal surfaces in \({\mathbb{R}^{\sc N},}\) \({\forall\,{\sc N} >3 .}\)

Mathematics Subject Classification

53C43 53C42 30F15 49Q05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlfors L.V.: The theory of meromorphic curves. Acta Soc. Sci. Fennicae. Nova Ser. A 3, 1–31 (1941)MathSciNetGoogle Scholar
  2. 2.
    Alarcón A., Fernández I.: Complete minimal surfaces in \({\mathbb {R}^3}\) with a prescribed coordinate function. Differ. Geom. Appl. 29(1 suppl), S9–S15 (2011)zbMATHCrossRefGoogle Scholar
  3. 3.
    Alarcón, A., Fernández, I., López, F.J.: Complete minimal surfaces and harmonic functions. Comment. Math. Helv. (2010, in press)Google Scholar
  4. 4.
    Alarcón A., López F.J.: Minimal surfaces in \({\mathbb {R}^3}\) properly projecting into \({\mathbb {R}^2}\) . J. Differ. Geom. (2012, in press)Google Scholar
  5. 5.
    Alarcón A., López F.J.: Null curves in \({\mathbb {C}^3}\) and Calabi-Yau conjectures. Math. Ann. (2012, in press)Google Scholar
  6. 6.
    Chern S.S.: Minimal Surfaces in an Euclidean Space of N Dimensions. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 187–198. Princeton Univ. Press, Princeton (1965)Google Scholar
  7. 7.
    Chern S.S., Osserman R.: Complete minimal surfaces in euclidean n-space. J. Anal. Math. 19, 15–34 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Colding T.H., Minicozzi W.P.: The Calabi-Yau conjectures for embedded surfaces. Ann. Math. 167(2), 211–243 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fujimoto H.: Extensions of the big Picard’s theorem. Tohoku Math. J. 24, 415–422 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fujimoto H.: On the Gauss map of a complete minimal surface in \({\mathbb {R}^{m} }\) . J. Math. Soc. Jpn. 35, 279–288 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Fujimoto H.: Modified defect relations for the Gauss map of minimal surfaces. II. J. Differ. Geom. 31, 365–385 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fujimoto H.: Examples of complete minimal surfaces in \({\mathbb {R}^m}\) whose Gauss maps omit m(m + 1)/2 hyperplanes in general position. Sci. Rep. Kanazawa Univ. 33, 37–43 (1988)MathSciNetGoogle Scholar
  13. 13.
    Jones P.W.: A complete bounded complex submanifold of \({\mathbb {C}^3}\) . Proc. Am. Math. Soc. 76, 305–306 (1979)zbMATHGoogle Scholar
  14. 14.
    Jorge L.P.M., Xavier F.: A complete minimal surface in \({\mathbb {R}^3}\) between two parallel planes. Ann. Math. 112(2), 203–206 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Klotz Milnor T.: Mapping surfaces harmonically into E n. Proc. Am. Math. Soc. 78, 269–275 (1980)zbMATHGoogle Scholar
  16. 16.
    Meeks III, W.H., Pérez, J., Ros A.: The embedded Calabi-Yau conjectures for finite genus (Preprint)Google Scholar
  17. 17.
    Osserman R.: Global properties of minimal surfaces in E 3 and E n. Ann. Math. 80(2), 340–364 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Osserman, R.: A Survey of Minimal Surfaces. Second edition. Dover Publications, Inc., New York, vi+207 pp. (1986)Google Scholar
  19. 19.
    Ru M.: On the Gauss map of minimal surfaces immersed in \({\mathbb {R}^n}\) . J. Differ. Geom. 34, 411–423 (1991)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wu, H.: The Equidistribution Theory of Holomorphic Curves. Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1970)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Antonio Alarcón
    • 1
  • Isabel Fernández
    • 2
  • Francisco J. López
    • 1
  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Departamento de Matemática Aplicada IUniversidad de SevillaSevillaSpain

Personalised recommendations