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Symmetry Aspects of 3-Periodic Minimal Surfaces

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Bifurcation and Symmetry

Abstract

Symmetry properties of 3-periodic minimal surfaces subdividing R3 into two congruent regions are discussed. The relation between the order of a flat point and its site symmetry is established. Explicit formulae are given for the calculation of the genus of such a surface depending on the kind of surface patches that build up the surface. Making use of 2-fold axes that have to be embedded in a surface with given symmetry new families of minimal balance surfaces have been derived. Two examples of bifurcations related to minimal surfaces are mentioned.

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References

  1. W. Fischer & E. Koch, Z. Kristallogr. 179, 31–52 (1987).

    Article  Google Scholar 

  2. A. L. Mackay & J. Klinowski, Comp. & Maths. with Appls. 12B, 803–824 (1986).

    Google Scholar 

  3. E. Koch & W. Fischer, Z. Kristallogr. 183, 129–152 (1988).

    Google Scholar 

  4. H. A. Schwarz, Gesammelte Abhandlungen, Band 1, Springer-Verlag, Berlin (1890).

    Book  Google Scholar 

  5. A. H. Schoen, NASA Technical Note D-5541 (1970).

    Google Scholar 

  6. W. Fischer & E. Koch, Acta Crystallogr. A45, 166–169 (1989).

    Google Scholar 

  7. J. C. C. Nitsche, Coll. Phys., Tome 51, Colloque C7, 265–271 (1990).

    Google Scholar 

  8. E. Koch & W. Fischer, Acta Crystallogr. A45, 169–174 (1989).

    Google Scholar 

  9. H. Karcher, Manuscripta Math. 64, 291–357 (1989).

    Article  Google Scholar 

  10. W. Fischer & E. Koch, Acta Crystallogr. A45, 485–490 (1989).

    Google Scholar 

  11. E. Koch & W. Fischer, Acta Crystallogr. A45, 558–563 (1989).

    Google Scholar 

  12. W. Fischer & E. Koch, Acta Crystallogr. A45, 726–732 (1989).

    Google Scholar 

  13. E. Koch & W. Fischer, Acta Crystallogr. A46, 33–40 (1990).

    Google Scholar 

  14. W. Fischer & E. Koch, Coll. Phys., Tome 51, Colloque C7, 131–147 (1990).

    Google Scholar 

  15. S. T. Hyde, Z. Kristallogr. 187, 165–185 (1989).

    Article  Google Scholar 

  16. H. Hopf, Springer Lecture Notes in Mathematics No. 1000 (1983).

    Google Scholar 

  17. J. C. C. Nitsche, Arch. Rat. Mech. Anal. 30, 1–11 (1968).

    Google Scholar 

  18. J. C. C. Nitsche, Ann. Acad. Sci. Fenn. AI 2, 361–373 (1976).

    Google Scholar 

  19. J. C. C. Nitsche, Lectures on minimal surfaces, Vol. 1, Cambridge Univ. Press, Cambridge (1989).

    Google Scholar 

  20. J. C. C. Nitsche, Arch. Rat. Mech. Anal. 52, 319–329 (1973).

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. (und Wissenschaftliches Zentrum für Materialwissenschaften), Institut für Mineralogie, Petrologie und Kristallographie, Hans-Meerwein-Straße, D-3550, Marburg, Germany

    Werner Fischer & Elke Koch

Authors
  1. Werner Fischer
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  2. Elke Koch
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Editor information

Editors and Affiliations

  1. Dept. of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    Eugene L. Allgower

  2. Fachbereich Mathematik, Universität Marburg, Hans-Meerwein-Str., Lahnberge, D-W-3550, Marburg, Germany

    Klaus Böhmer

  3. Dept. of Mathematics, University of Houston, University Park, Houston, TX, 77204-3476, USA

    Martin Golubitsky

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© 1992 Birkhäuser Verlag Basel

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Fischer, W., Koch, E. (1992). Symmetry Aspects of 3-Periodic Minimal Surfaces. In: Allgower, E.L., Böhmer, K., Golubitsky, M. (eds) Bifurcation and Symmetry. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 104. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7536-3_11

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  • DOI: https://doi.org/10.1007/978-3-0348-7536-3_11

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-7538-7

  • Online ISBN: 978-3-0348-7536-3

  • eBook Packages: Springer Book Archive

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