The gyroid is embedded and has constant mean curvature companions

  • Karsten Große-Brauckmann
  • Meinhard Wohlgemuth 


The gyroid is a triply periodic minimal surface in the associated family of the SchwarzP- andD-surface. We prove it is embedded and find constant mean curvature companions of the gyroid with small constant mean curvature. We also discuss a surface similar to the gyroid in the associated family of the SchwarzH-surface.


System Theory Minimal Surface Curvature Companion Periodic Minimal Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ADNS]
    Anderson, D.M., H.T. Davis, J.C.C. Nitsche, L.E. Scriven: Periodic surfaces of prescribed mean curvature. Adv. Chem. Phys.77, Wiley, New York 1990Google Scholar
  2. [BC]
    Barbosa, J.L., M. do Carmo: On the size of a stable minimal surface in ℝ3. Am. J. Math.98, 515–528 (1976)Google Scholar
  3. [DHKW]
    Dierkes, U., Hildebrandt, S., Küister, A., Wohlrab, O.: Minimal surfaces. Springer Berlin, 1993Google Scholar
  4. [DP]
    E. Dubois-Violette, B. Pansu (Editors): International workshop on geometry and interfaces, Aussois, France, Sept. 1990. Journal de Physique, Colloque, C7, supplement au No. 23, Tome 51Google Scholar
  5. [FKZ]
    Förster, S., A.K. Khandpur, J. Zhao, F. Bates, I.W. Hamley, AJ. Ryan, W. Bras: Complex phase behavior of polyisoprene-polystyrene diblock copolymers near the order-disorder transition. Macromolecules27, 6922–6935 (1994)Google Scholar
  6. [GT]
    Gilbarg, D., N.S. Trudinger: Elliptic partial differential equations of second order, Second edition, Springer Berlin, 1983Google Scholar
  7. [G1]
    Große-Brauckmann, K.: New surfaces of constant mean curvature. Math. Z.214, 527–565 (1993)Google Scholar
  8. [G2]
    Große-Brauckmann, K.: The family of constant mean curvature gyroids. Gang Preprint IV.13, University of Amherst (1995)Google Scholar
  9. [G3]
    Große-Brauckmann, K.: Stable constant mean curvature surfaces minimize area. Pac. J. (to appear)Google Scholar
  10. [HGT]
    Hajduk, D.A., P.E. Harper, S.M. Gruner, C.C. Honecker, G. Kim, E.L. Thomas, L.J. Fetters: The gyroid: A new equilibrium morphology in weakly segregated diblock copolymers. Macromolecules27, 4063–4075 (1994)Google Scholar
  11. [Kp]
    Kapouleas, N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math.131, 239–330 (1990)Google Scholar
  12. [K]
    Karcher, H.: The triply periodic minimal surfaces of A. Schoen and their constant mean curvature companions. Manus. math.64, 291–357 (1989)Google Scholar
  13. [LL]
    Lidin, S., S. Larsson: Bonnet transformation of infinite periodic minimal surfaces with hexagonal symmetry. J. Chem. Soc. Faraday Trans.86, 769–775 (1990)Google Scholar
  14. [MW]
    Meeks, W.H., B. White: Minimal surfaces bounded by convex curves in parallel planes. Comm. Math. Helv.66, 263–278 (1991)Google Scholar
  15. [N]
    Nitsche, J.C.C.: Lectures on minimal surfaces I. Cambridge University Press, 1989Google Scholar
  16. [O]
    Osserman, R.: A survey of minimal surfaces, second edition, Dover, New York 1986Google Scholar
  17. [R]
    Ross, M.: Schwarz'P andD surfaces are stable. Differ. Geom. Appl.2, 179–195 (1992)Google Scholar
  18. [S]
    Schoen, A.H.: Infinite periodic minimal surfaces without selfintersections. NASA Technical Note TN D-5541 (1970)Google Scholar
  19. [Sch]
    Schoen, R.: Estimates for stable minimal surfaces in three dimensional manifolds. In: Seminar on Minimal Submanifolds. Ann. Math. Stud.103. Princeton: 1983Google Scholar
  20. [TAHH]
    Thomas, E.L., D.M. Anderson, C.S. Henkee, D. Hoffman: Periodic area-minimizing surfaces in block copolymers. Nature334, 598–601 (1988)Google Scholar
  21. [W]
    White, B.: The space of m-dimensional surfaces that are stationary for a parametric elliptic functional. Indiana Univ. Math. J.36, 567–603 (1987)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Karsten Große-Brauckmann
    • 1
  • Meinhard Wohlgemuth 
    • 1
  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

Personalised recommendations