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Complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order

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Abstract

Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstraß Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Choe J., Soret M.: Nonexistence of certain complete minimal surfaces with planar ends. Comment. Math. Helv. 75, 189–199 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fang Y., Hwang J.F.: A note on Shiffman’s theorems. Geom. Dedicata 81(1–4), 167–171 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hoffman, D., Karcher, H.: Complete embedded minimal surfaces of finite total curvature. In: Osserman, R. (ed.) “Geometry V”, Encyclopedia of Math. Sci., vol. 90, pp. 5–93. Springer (1997)

  5. Hoffman D., Meeks W.H. III: The strong halfspace theorem for minimal surfaces. Invent. Math. 101, 373–375 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Meeks W.H. III, Rosenberg H.: The maximum principle at infinity for minimal surfaces in flat three manifolds. Comment. Math. Helv. 65, 255–270 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Obreschkoff, N.: Verteilung und Berechnung der Nullstellen reeller Polynome. VEB Deutscher Verlag der Wissenschaften (1963)

  8. Osserman R.: A Survey of Minimal Surfaces. Dover Publications, New York (1986) 2. Aufl.

    Google Scholar 

  9. Pérez J., Ros A.: Properly embedded minimal surfaces of finite total curvature. Lect. Notes Math. 1775, 15–66 (2001)

    Google Scholar 

  10. Pérez J., Ros A.: Some uniqueness and nonexistence theorems for embedded minimal surfaces. Math. Ann. 295(3), 513–525 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. Traizet, M.: Exploring the space of embedded minimal surfaces of finite total curvature. Preprint (2007)

  14. Zimmermann, R.: Vollständige eingebettete Minimalflächen endlicher Totalkrümmung mit planaren Enden kleinster Ordnung, Dissertation, Kiel (2008). http://eldiss.uni-kiel.de/macau/receive/dissertation_diss_00002937

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  1. Ralf Zimmermann

    Present address: German Aerospace Center, Lilienthalplatz 7, 38108, Brunswick, Germany

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  1. Christian-Albrechts-Universität Kiel, Ludewig-Meyn Str. 4, 24098, Kiel, Germany

    Ralf Zimmermann

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  1. Ralf Zimmermann
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Correspondence to Ralf Zimmermann.

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Zimmermann, R. Complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order. Math. Ann. 346, 85–105 (2010). https://doi.org/10.1007/s00208-009-0390-0

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  • DOI: https://doi.org/10.1007/s00208-009-0390-0

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