Skip to main content
SpringerLink
Log in

The embedded singly periodic Scherk-Costa surfaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of with finite total curvature.

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. Bredon, G.E.: Topology and Geometry. Grad. Text in Math. Springer Verlag, New York 139, (1993)

  2. Costa, C.: Example of a complete minimal immersion in of genus one and three embedded ends. Bol. Soc. Bras. Mat. 15, 41–54 (1984)

    Google Scholar 

  3. Cheng, S.Y.: Eigenfunctions and Nodal Sets. Comment. Math. Helvetici. 51, 43–55 (1976)

    MATH  Google Scholar 

  4. Ferrer, L., Martín, F.: Minimal surfaces with helicoidal ends. Mathematische Zeitschrift, posted on April 15, 2005, DOI: 10.1007/s00209-005-0777-x (to appear in print)

  5. Hoffman, D., Karcher, H.: Complete embedded minimal surfaces of finite total curvature. Encyclopedia of Math. Sci., Springer Verlag 90, 5–93 (1997)

    MATH  Google Scholar 

  6. Hoffman, D., Karcher, H., Wei, F.: The singly periodic genus-one helicoid. Comment. Math. Helv. 74, 248–279 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hoffman, D., Meeks, W.: Embedded minimal surfaces of finite topology. Ann. Math. 131, 1–34 (1990)

    Article  MATH  Google Scholar 

  8. Karcher, H.: Construction of minimal surfaces. Surveys in Geometry, University of Tokyo 1–96 (1989) and Lecture Notes 12, SFB256, Bonn (1989)

  9. López, F.J., Martín, F.: Complete minimal surfaces in . Publicacions Matemàtiques 43, 341–449 (1999)

    MATH  Google Scholar 

  10. Meeks, W.H., Rosenberg, H.: The global theory of doubly periodic minimal surfaces. Invent. Math. 97, 351–379 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nitsche, J.C.C.: Lectures on minimal surfaces. Cambridge University Press, Cambridge, (1989)

  12. Osserman, R.: A survey of minimal surfaces. Dover, New York, 2nd ed (1986)

  13. Riemann, B.: Über die Fläche vom Kleinsten Inhalt bei gegebener Begrenzung. Abh. Königl. Ges. d. Wiss. Göttingen, Mathem. Cl. 13, 3–52 (1867)

    Google Scholar 

  14. Ramos Batista, V.: Construction of new complete minimal surfaces in based on the Costa surface. Doctoral thesis, University of Bonn, Germany (2000)

  15. Ramos Batista, V.: Accurate evaluation of elliptic integrals. Technical Report 61/03, University of Campinas (2003) http://www.ime.unicamp.br/rel_pesq/2003/rp61-03.html

  16. Ramos Batista, V.: Polynomial estimates in the unitary interval. Technical Report 60/03, University of Campinas (2003) http://www.ime.unicamp.br/rel_pesq/2003/rp60-03.html

  17. Ramos Batista, V.: Singly periodic costa surfaces. Journal London math. soc. 72, 478-498 (2005)

    Article  MATH  Google Scholar 

  18. Scherk, H.F.: Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen. Journal für die reine und angewandte Mathematik 13, 185–208 (1835)

    Article  MATH  Google Scholar 

  19. Schoen, R.: Uniqueness, symmetry and embeddedness of minimal surfaces. J. Differential Geom. 18, 791–809 (1983)

    MATH  MathSciNet  Google Scholar 

  20. Schwarz, H.A.: Gesammelte Mathematische Abhandlungen. Springer, Berlin (1890)

  21. Traizet, M.: Construction de surfaces minimales en recollant des surfaces de Scherk. Ann. Inst. Fourier, Grenoble, 46, 1385-1442 (1996)

    MATH  MathSciNet  Google Scholar 

  22. Traizet, M.: An embedded minimal surface with no symmetries. J. Differential Geom. 60, 103–153 (2002)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Depto. Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Francisco Martín

  2. IMECC-Unicamp, P.O.BOX 6065, 13083-970, Campinas, Brazil

    Valério Ramos Batista

Authors
  1. Francisco Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valério Ramos Batista
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valério Ramos Batista.

Additional information

Dedicated to Hermann Karcher on the Occasion of his 65th Birthday

Martín's research is partially supported by MCYT-FEDER grant number BMF2001-3489.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martín, F., Batista, V. The embedded singly periodic Scherk-Costa surfaces. Math. Ann. 336, 155–189 (2006). https://doi.org/10.1007/s00208-006-0778-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-006-0778-z

Keywords

Search

Navigation