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A characterisation of the Hoffman–Wohlgemuth surfaces in terms of their symmetries

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Abstract

For an embedded singly periodic minimal surface \({\tilde{M}}\) with genus \({\varrho\ge4}\) and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman–Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces.

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References

  1. Callahan M., Hoffman D., Meeks W.H.: Embedded minimal surfaces with an infinite number of ends. Invent. Math. 96, 459–505 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Callahan M., Hoffman D., Meeks W.H.: The structure of singly-periodic minimal surfaces. Invent. Math. 99, 455–481 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Callahan M., Hoffman D., Karcher H.: A family of singly periodic minimal surfaces invariant under a screw motion. Exp. Math. 2, 157–182 (1993)

    MATH  MathSciNet  Google Scholar 

  4. Choi H.I., Meeks W.H., White B.: A rigidity theorem for properly embedded minimal surfaces in \({\mathbb{R}^3}\) . J. Differ. Geom. 32, 65–76 (1990)

    MATH  MathSciNet  Google Scholar 

  5. Costa C.J.: Uniqueness of minimal surfaces embedded in \({\mathbb{R}^3}\) with total curvature 12π. J. Differ. Geom. 30, 597–618 (1989)

    MATH  Google Scholar 

  6. Ferrer L., Martín F.: Minimal surfaces with helicoidal ends. Math. Z 250, 807–839 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Kapouleas N.: Complete embedded minimal surfaces of finite total curvature. J. Differ. Geom. 47, 95–169 (1997)

    MATH  MathSciNet  Google Scholar 

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

  12. Lazard-Holly L., Meeks W.H.: Classification of doubly-periodic minimal surfaces of genus zero. Invent. Math. 143, 1–27 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. López F.J., Martín F.: Complete minimal surfaces in \({\mathbb{R}^3}\) . Publ. Mat. 43, 341–449 (1999)

    MATH  MathSciNet  Google Scholar 

  14. López F.J., Ros A.: On embedded complete minimal surfaces of genus zero. J. Differ. Geom. 33, 293–300 (1991)

    MATH  Google Scholar 

  15. Martín F.: A note on the uniqueness of the periodic Callahan-Hoffman-Meeks surfaces in terms of their symmetries. Geom. Dedicata 86, 185–190 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Martín F., Rodríguez D.: A characterization of the periodic Callahan-Hoffman-Meeks surfaces in terms of their symmetries. Duke Math. J. 89, 445–463 (1997)

    Article  MathSciNet  Google Scholar 

  17. Martín F., Weber M.: On properly embedded minimal surfaces with three ends. Duke Math. J. 107, 533–559 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Massey, W.S.: Algebraic topology: an introduction. In: Graduate Texts in Mathematics Springer, New York

  19. Meeks W.H., Rosenberg H.: The uniqueness of the helicoid. Ann. Math. 161, 727–758 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Meeks W.H., Pérez J., Ros A.: Uniqueness of the Riemann minimal examples. Invent. Math. 131, 107–132 (1998)

    Article  Google Scholar 

  21. Nitsche J.C.C.: Lectures on Minimal Surfaces. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  22. Osserman, R.: The convex hull property of immersed manifolds. J. Differ. Geom. 6, 267–270 (1971/72)

    Google Scholar 

  23. Osserman R.: A Survey of Minimal Surfaces. Dover, New York (1986)

    Google Scholar 

  24. Pérez J., Traizet M.: The classification of singly periodic minimal surfaces with genus zero and Scherk type ends. Trans. Am. Math. Soc. 359, 965–990 (2007)

    Article  MATH  Google Scholar 

  25. Pérez J., Rodríguez M., Traizet M.: The classification of doubly periodic minimal tori with parallel ends. J. Differ. Geom. 69, 523–577 (2005)

    MATH  Google Scholar 

  26. Ramos Batista V.: A family of triply periodic costa surfaces. Pac. J. Math. 212, 347–370 (2003)

    Article  MathSciNet  Google Scholar 

  27. Ramos Batista V.: Noncongruent minimal surfaces with the same symmetries and conformal structure. Tohoku Math. J 56, 237–254 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schoen R.: Uniqueness, symmetry and embeddedness of minimal surfaces. J. Differ. Geom. 18, 701–809 (1983)

    Google Scholar 

  29. Traizet M.: Adding handles to Riemann’s minimal surfaces. J. Inst. Math. Jussieu 1(1), 145–174 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  31. Weber M.: A Teichmüller theoretical construction of high genus singly periodic minimal surfaces invariant under a translation. Manuscr. Math. 101, 125–142 (2000)

    Article  MATH  Google Scholar 

  32. Wohlgemuth M.: Minimal surfaces of higher genus with finite total curvature. Arch. Ration. Mech. Anal. 137, 1–25 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CMCC—ABC Federal University, r. Catequese 242, 3rd floor, Santo André, SP, 09090-400, Brazil

    V. Ramos Batista

  2. IME—University of São Paulo, r. do Matão 1010, São Paulo, SP, 05508-090, Brazil

    P. Simões

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  1. V. Ramos Batista
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  2. P. Simões
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Correspondence to V. Ramos Batista.

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Ramos Batista, V., Simões, P. A characterisation of the Hoffman–Wohlgemuth surfaces in terms of their symmetries. Geom Dedicata 142, 191–214 (2009). https://doi.org/10.1007/s10711-009-9366-1

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  • DOI: https://doi.org/10.1007/s10711-009-9366-1

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