On limits of triply periodic minimal surfaces

Article
  • 21 Downloads

Abstract

In this paper, we introduce generic limits of triply periodic minimal surfaces and consider the genus-three case. We will prove that generic limits of such minimal surfaces consist of a one-parameter family of Karcher’s saddle towers and Rodríguez’ standard examples.

Keywords

Minimal surface Triply periodic limit 

Mathematics Subject Classification

Primary 53A10 Secondary 49Q05 53C42 

References

  1. 1.
    Ejiri, N., Fujimori, S., Shoda, T.: A remark on limits of triply periodic minimal surfaces of genus 3. Topol. Appl. 196(part B), 880–903 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Farkas, H.M., Kra, I.: Riemann Surfaces, Graduate Texts in Mathematics, vol. 71, 2nd edn. Springer, New York (1992)CrossRefGoogle Scholar
  3. 3.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry, Reprint of the 1978 Original. Wiley Classics Library. Wiley, New York (1994)Google Scholar
  4. 4.
    Hoffman, D., Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101, 373–377 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hoffman, D., Wei, F., Karcher, H.: The genus one Helicoid and the minimal surfaces that led to its discovery, Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), pp. 119–170. Publish or Perish, Houston, TX (1993)Google Scholar
  6. 6.
    Karcher, H.: Embedded minimal surfaces derived from Scherk’s examples. Manuscr. Math. 62(2), 83–114 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lazard-Holly, H., Meeks III, W.H.: Classification of doubly-periodic minimal surfaces of genus zero. Invent. Math. 143(1), 1–27 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lidin, S., Larsson, S.: Bonnet transformation of infinite periodic minimal surfaces with hexagonal symmetry. J. Chem. Soc. Farday Trans. 86(5), 769–775 (1990)CrossRefGoogle Scholar
  9. 9.
    Meeks III, W.H.: The theory of triply periodic minimal surfaces. Indiana Univ. Math. J. 39(3), 877–936 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Meeks III, W.H., Rosenberg, H.: The global theory of doubly periodic minimal surfaces. Invent. Math. 97(2), 351–379 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Meeks III, W.H., Rosenberg, H.: The geometry of periodic minimal surfaces. Comment. Math. Helv. 68(4), 538–578 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Meeks III, W.H., Rosenberg, H.: The uniqueness of the helicoid. Ann. Math. (2) 161(2), 727–758 (2005)Google Scholar
  13. 13.
    Nagano, T., Smyth, B.: Periodic minimal surfaces. Comment. Math. Helv. 53(1), 29–55 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pérez, J., Rodríguez, M.M., Traizet, M.: The classification of doubly periodic minimal tori with parallel ends. J. Differ. Geom. 69(3), 523–577 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pérez, J., Traizet, M.: The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends. Trans. Am. Math. Soc. 359(3), 965–990 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rodríguez, M.M.: The space of doubly periodic minimal tori with parallel ends: standard examples. Mich. Math. J. 55(1), 103–122 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ros, A.: One-sided complete stable minimal surfaces. J. Differ. Geom. 74(1), 69–92 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Schröder-Turk, G.E., Fogden, A., Hyde, S.T.: Bicontinuous geometries and molecular self-assembly: comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces. Eur. Phys. J. B 54, 509–524 (2006)CrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMeijo UniversityTempaku, NagoyaJapan
  2. 2.Department of MathematicsOkayama UniversityTsushimanaka, OkayamaJapan
  3. 3.Faculty of EducationSaga UniversitySaga-cityJapan

Personalised recommendations